I'd recommend the first year do them through something like taxact. then print out your forms from that, and do them again by hand just to make sure you understand it and get the numbers right. next year just do them yourself.
I'd actually reverse the order - do it by hand first, then run the software. If you run the tax software first, it's easy to fall into the trap of thinking that the tax software is 100% correct and optimal. Your tax software can become a crutch. If you start with pencil and paper first, you're more likely to gain a better understanding of the tax code. Also, it is possible, though highly unlikely, that there is a bug in the software.
Also, I should note that there exist situations where minimizing your federal taxes can increase your state taxes. I encountered this myself as a student (it involved a fellowship and the lifetime learning credit - I could explain if somebody would like, but the details are not all that meaningful to this discussion). Some tax software may try to optimize your overall tax bill of federal + state, but some may not. I know for a fact that when I tried H&R Block online, it did not do this, as it would refuse to let me fill out my state tax forms without first submitting the federal forms to the IRS.
Start with federal first for two reasons:
1) Most state returns rely on you having figured out your AGI from your federal
2) Having lived in multiple states in one year can be tricky.