My salary is doubling next week when I start a new job (I deliberately took a low salary to work 20 hours a week). Could this cause any income tax issues? For example, did I not withhold at a high enough rate for the first few months of the year and will I need to pay a penalty?
Assuming you are a normal, W-2 employee, with taxes withheld, you will likely end up withholding more than needed. Your employer will withhold taxes as if you earned your new, higher, salary for the entire year. But you didn't. You made substantially less for a few months.
Example: Let's say your initial salary kept you in the 15% tax bracket, but your new salary puts you solidly in the 25% bracket, if you made that for the entire year. So your employer will withhold at a rate as if a portion of your salary is in the 25% tax bracket. However, it's possible that your total annual salary still is in the 15% tax bracket, so none of your income should be withheld at the 25% rate.
I use the extreme example of switching tax brackets in an effort to make it a little bit clearer. However, this is true at all income levels >$9,275 (single), because of the progressive nature of the tax system.
A similar situation happened to me a few years ago when I graduated. My income nearly tripled for the last 7 months of the year. Got a decent refund of a few hundred dollars. Only time in the past 5 or 6 years at least that I actually got a refund.
TL;DR -- it's not that you didn't withhold enough for the first few months, you're withholding too much for the rest of the year. You'll be fine.
You won't have to pay a penalty just for withholding too little the first part of the year. You can increase your withholding a little to make up for the difference if you like to avoid a big tax bill or refund, or wait a few months and estimate how much your tax will be for the year, and adjust as necessary.
You'll only need to pay a penalty if at the end of the year you have withheld less than 90% of your total tax bill for the year, or your bill minus $1000, or 100% of last year's tax bill (110% if AGI was over $150k, but from your Q that sounds unlikely), whichever is least.
So if you are making more this year than you did last year, so long as you withhold more this year than your total tax last year you'll be fine.
Are you working as a W-2 employee or as a 1099 Contractor? If W-2 with taxes withheld, you should be fine not needing to catch up per se. Depends upon how much your total annual wages are for 2017 as to whether you'll owe or not - and, how many exemptions you're claiming for withholding? Lots of factors jgoe...