Justin Cave has a good answer and Pete B.'s is decent, but there's a bit more to this that hasn't been said yet.
As Justin Cave mentioned, there's a lot of mortgage calculators out there that'll help you figure out some of the pure math of the mortgage, but that's only skimming the surface of what you'll need to know about the process, so I won't go into that.
Pete suggests to get rid of the student loans and pay off a little of the car loan. The problem with that is student loans are considered very differently on credit than other debt. Car loans are secured debt, until they aren't. If you get in a car wreck and it's totaled, your insurance will pay some (if you aren't at fault), but likely not all, of the loan. The remaining debt becomes unsecured, so it's not a good kind of debt to have when looking at a mortgage. Mortgage lenders know this, and will handle it accordingly.
Student loans aren't secured debt and they aren't unsecured debt. Like medical bills, they are in a class by themselves. They affect your credit less, good and bad, so are a better thing to have than other debt. Miss a couple of payments on your car, and your credit can drop substantially. Miss a couple of payments on your student loan and your student loan company has many, legally defined ways to work with you to keep you from screwing yourself and for the student loan company from screwing you. Other loans don't have these protections. And your credit isn't as drastically affected.
I'd suggest paying off the car before you pay off your student loans. I'd also suggest keeping at least 6 months of your after-mortgage expenses for an emergency fund. As an off-the-cuff number, I'd suggest dropping $20k of your savings on your car loan, while keeping the rest for your emergency fund.
You don't mention credit card debt, but I'm going to assume you have some. That's definitely unsecured debt, so pay that off even before paying off the car.
Besides looking better on your mortgage application, credit card debt can have a drastic negative effect on your credit score. Getting rid of it avoids getting hit multiple times on your application for debt. Yes, it's not supposed to affect you multiple times, but it does. It affects your credit score, your debt to income ratio, and having lots of credit card debt makes it look like you are willing to continue to increase your credit card debt beyond what's financially feasible for you. Mortgage lenders will tell you they don't think like this, but they do even if it's unconsciously. It may even be part of their company policy.
At 23, you likely don't have good credit. This is simply because of the short time you've had credit. This isn't "ageism", it's just a fact. I'm in my early 40's and my credit is negatively impacted because of my relative short credit history. Your credit history is a combination of what credit you've had in the past and paid off, as well as what credit you are currently still paying on. When I got my mortgage earlier this year, I only had a car loan of 2-3 years on my report, so I was hampered by that. I have a whole long 20+ year history of student and personal loans that've been paid off, but that didn't work for me as much as the short car loan worked against me.
If I would do it again, I'd actually keep a couple hundred on a few of the student loans and pay off the car, similar to what I advised you earlier. I'd had the student loans so long and they had dogged me for so long when I was under and unemployed that I just wanted them gone, so I didn't think about the credit implications. I also didn't know the full credit implications until afterwards, though.
Because you are just out of college, you'll also likely have to give the broker a copy of your $70k job offer as proof of your potential earnings, if you aren't working at the job by then. They get tons of people who try to tell them how much they are going to be making, only to not get the job they were hoping for.
Another thing to think about is that you'll likely need to meet a minimum steady job requirement. I had to wait a few months to get my loan approved, since I had left a job to move 2000 miles and had trouble finding another job at the new location (long story). My mortgage company only needed 1 year steady employment before they would give me a loan. Some companies need 2 or more years. This doesn't mean staying at one job through all that, but it does mean having a job with little to no gaps during that time. They may also have an upper limit built into the job requirement too, so that you aren't job hopping every month.
If you don't already have an account, go to Credit Karma's website to find out approximately what your credit history looks like. I'm not affiliated with them in any way, I just know it was indispensable to know what to expect when I was looking at a mortgage or even a car loan. They give you Experian and TransUnion credit scores, as well as access to a lot of other information about your credit, including history. It also keeps track of your credit scores, as long as you keep logging in regularly. It'll help you figure out what's helping and hurting your score. The site also includes a lot of good generic and general advice to help. Aaaaand it includes a bunch of ads for credit cards and other junk. I'd stay away from all that and just stick to the free info.
There are other sites that will do something similar, but this is the one I've used. I can't comment on how any other site works.
You checking your history won't affect your score. It used to, a handful of years ago, but things changed and now there are "soft" and "hard" hits on your credit.
A soft hit is you checking your credit, a credit card company's prequalification credit check, or a loan officer doing an initial query on your score. These soft hits don't affect your credit score at all. You could check your score 5 times a week, and it wouldn't matter.
A hard hit is what a loan officer does when they officially get your full credit history for the purposes of giving you a loan. You can do this 1-2 times with a mortgage or 3-5 times with a car loan before they start affecting your credit. They fall off your credit history after 12-15 months, as opposed to the 7 years most other things take. These hard hits usually don't affect your credit much, unless you have a very large number of them. Just 1-2 won't affect your score, but 3-5 will take 1-5 points off and more than that will start costing you even more points.
As a US Army Reserve Veteran, I was able to get a VA loan. This allowed me to get 100% financing as well as a few other (minor) perks. I still overpaid my first month by quite a bit in lieu of a down payment, but that wasn't required by the lender.
Getting a VA loan requires more paperwork to verify your status as a Vet, and you'll need something other than a Dishonorable Discharge to get into the program, but it's worth it. Not only does it save you money, but the VA office will likely hook you up with a realtor that understands the VA loan process and can make all of it easier. I got a wonderful realtor that explained things to me, including telling me things I should be asking, things to avoid agreeing to, and so much more I didn't know about as a first time home buyer.
You don't mention being a Veteran, but I thought I would just in case and for others reading this Answer. I also mentioned it in a comment on Justin Cave's Answer, but I thought I'd also mention it here.
There's a point system when it comes to mortgages. It affects your interest rate and your monthly payment. I don't fully understand it, but your credit, down payment, and mortgage broker may all affect those points, which will have long term effects on your mortgage. I don't understand it all, so that's all I'll say about it. I just wanted you to know about it, since it was a surprise to me and is something to ask about when you talk to a mortgage broker.
They can be your best friend and your worst enemy, and at the same time. Fortunately, I got a good one that worked with me and helped me get over some obstacles. It's their job to tell you sometimes bad news. Try not to take it personally, even if it hurts. There were things my broker said that surprised me, so just try to understand that they are working on a business deal that can cost you and them a lot of time and money, especially if something goes wrong. I'm not saying that they won't try to put you through a hard time or try to take advantage of you, but most are trying to make sure they don't have to foreclose on you.
These people will also let you know what your closing costs are as well as a dozen other things. Your realtor will help with some of this, including what the seller is willing to pay. This can get complicated quickly, so don't assume that an online mortgage calculator will tell you everything you need to save for.
I forgot to mention that with a typical mortgage, you're looking at 10-20% for a down payment, although you might qualify for a first time buyer loan and only need 3%.
As I mentioned in a comment on Justin Cave's Answer (I wasn't intending to write an answer when I made those comments), you're probably not going to be getting a mortgage very soon. I'd guess at least 6 months, just based on your saving's plan. You may have to wait even longer, based on employment requirements and what job(s) you've had in the past.
The housing market can go through a huge change in that short time, especially with the current pandemic. The market may bottom out and you get a great deal on a house with great interest rate, or the prices and interest might be inflated to cover losses during the pandemic. So whatever specific information you get about your situation in the next few weeks could be drastically different than in a few months. And it could easily go either way.
Your job may even evaporate in the coming months. This pandemic has caused a lot of uncertainty, and with you being a "new hire", they may get rid of you first or the whole company may just shut down until the virus gets under control. You might get lucky and start working from home like some us, but that's not a guarantee, either.
Don't assume anything, just keep doing good things for your credit and your down payment. That's always good advice, even if the mortgage goalposts move beyond what you can handle. Eventually, it'll move the other way and then you'll be in a good position to get what you want. And yes, I closed on my house shortly after the start of this pandemic. In fact, I didn't even find my house until after my city locked down. I had the paperwork started before then, but I hadn't found a house I wanted.
I've written extensively about spending habits on another Question, so I won't add it here. This is pretty long as it is.
Paying off debt and living within means vs. long term planning
Realize that getting a house isn't a fun or easy thing. You'll likely be searching for paperwork for weeks on end, maybe more. I know it took me quite a while the first time I applied for a mortgage. That was about a year before I tried it again earlier this year. (Part of that long story I mentioned earlier.) Even though I had most of the paperwork located the first time, I still had to get a whole bunch more the 2nd time. Some of it was duplicate/updated info, such as insurance, bank account status, employment status, and probably several other things.
Also, realize that no matter how good a house you get, there's likely a bunch of repairs that need done. Some of these will be done, or at least paid for, by the seller. Your realtor should be able to help you figure this out. That said, don't put all of your savings into the down payment. I've spent several thousand dollars doing repairs, upgrades, cleaning, replacements, and more. About half of what I've done could be considered unnecessary, but they make the house more of a home to me.
Just understand that there'll be things that you want to change and spend money on when you get into the place, so save money for that time. And don't spend "too much" on that stuff. I've wore down my emergency fund because of getting too enthusiastic with my changes. I also have a dozen partially completed projects started and need to complete them before I start new ones, let alone spend more on future projects. I'm suggesting balance to your down payment and your savings for after closing on the house.