Utilities and cost of living vary from city to city but maybe not that much. For basic planning purposes you can probably figure to spend as much as you are now, maybe a little more. And adjust as needed when you get there. (And adjust if, for example, you're moving from a very low cost of living area or to a very high cost of living area.)
The cost of housing varies quite a bit from city to city, but you can do this research using Zillow, Craigslist, other places.
Now, on to moving itself. The cost of moving can vary hugely depending on how much stuff you have and how much work you want to do. On the cheap end, you can rent a U-Haul or one of those portable boxes that they plant outside your old house and move for you. You'll do all the packing/loading/unloading/unpacking yourself but it saves quite a bit of money. My family and I moved from Seattle to California last year using one of those portable box places and it ended up costing us ~$1400 including 30 days of storage at the destination while we looked for a place. We have a <1000 sq foot place with some furniture but not a huge amount and did all the packing/loading ourselves. If we had wanted full service where people come pack, load, unpack, etc, it could have been 2-3x that amount. (And if we had more stuff, it could have been a lot more expensive too. Try not to acquire too much stuff as you just end up having to move it around and take care of it all!) Your employer may cover moving expenses, ask about this when talking about job offers. Un-reimbursed moving expenses are tax-deductible in the US (even if you don't itemize).
Since you're just starting out, your best bet is to overestimate how much you think things will cost, then adjust as you arrive and settle in for a few months. Try to save as much as you can, but remember to have fun too. Hope this helps!