You can file with multiple states in TurboTax.
Start with the Federal return and enter all the W2s including the Lines 15 through 17 for all the states. If you just worked in the other states you will pay taxes in those states and get a credit for taxes paid in your home state. You should do the non-residential states first and your residential state last. This is to ensure you get the credits derived from the non-residential states to flow properly.
Check to see if any of the states have reciprocity with your home state. If you do have states with reciprocity check to see if that state withheld on your W2. If the state withheld and they do have reciprocity with your home state you will need to file a return to reclaim the amount withheld. If they did not withhold then you are good to go.
[Edited 03/14/2020 08:37 PDT]
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Well...that an get a bit expensive...but
A)) For states with an income tax
1) Online software can only do 5 states....and can e-file the 5 states too.
2) Desktop/Download software can do many sates, but only e-file 3. any more than 3 have to be printed and mailed in
1) You enter just one W-2 for each employer, but you must put in a separate W-2 for each different employer. TTX can take as many as you reasonably have (Maybe a 10 or 20 limit....I've never pushed it, so I don't know the real limit).
2) IF you worked for one employer in multiple states, you just have that one W-2 to enter, but for boxes 15-to-17, you will enter separate line for each state you worked in. The employer's W-2 should list the income for each state you worked in.
3) Depending on your home/resident state....what extra Non-Resident states software you need to buy may depend on what your state, and the other states actual are.
...........a) Some states don't have an income tax, so no tax return would be required for that "other " state, but it would still be taxed by your home state (if your home state has an income tax).
..............b) some states could have a reciprocal tax agreement with your state, and you don't need a tax return for that other state (with some exceptions).
.................c) Most other states you would have to file as a non-resident for income earned in that state.
4) You must prepare ALL of the Federal section first, every scrap. Then all the non-resident sates, then your home/resident state last. (with some preparation order exceptions if your home & work state is between CA,AZ,OR, IN & sometimes VA)
IF you actually moved your permanent state residence during the year...and more that two states are involved...then it gets messier.