Welcome! You're already off to a fantastic start.
For Mint/tracking spending, you might want to categorize reimbursable expenses as "Hide from budgets and trends." It is important to figure out where that $200/mo of "shopping/other" is going! You can also split charges into multiple categories, etc, to categorize all your spending appropriately.
You also might like the Bogleheads wiki
https://www.bogleheads.org/wiki/Getting_started - I found this site really useful for getting my plans together.
Specific Question(s):
1. Should I be contributing to roth or regular IRAs and 401ks? What are my limits for each?
The Roth vs. Traditional really comes down to two things - (A) What you expect your tax rate to be in retirement. If you expect lower taxes, a traditional would be better. If you expect higher taxes, a Roth would be better. If you expect the same tax rate, it's a wash. Most people on these forums are expecting lower tax rates, so traditional accounts would be my initial suggestion, without knowing more about your goals. (B) Flexibility - Roth accounts can have contributions taken out before age 59 1/2 without penalty (contributions only, not gains). Because of this, they could be used as emergency accounts, or as the first accounts you draw down on in early retirement.
The limits are $5,500 /year for IRAs and $18,000 for 401ks
2. How much cash should I keep in a bank account compared to investing?
This really depends on your risk tolerance, as well as more specific life factors - how stable is your job/income, for example. Even if your job is very stable, you might want to keep at least a few months living expenses on hand. $10k seems more than enough.
3. Once I max out my IRA and 401k for the year, where/how should I invest the rest of my non-cash money?
HSA, for sure
http://www.madfientist.com/ultimate-retirement-account/Then you would start taxable accounts, at Vanguard or other institutions.
4. I am frantically listening to podcasts and reading blogs, but are there other things I should look into doing immediately to save money?
Definitely try to get a better grasp of the spending with Mint, so you can see what makes the most sense to cut out. You could possibly lower phone costs
http://www.techmeshugana.com/, you could lower restaurant spending if you wanted, and you should at least figure out where exactly that $200/mo shopping/other is going.
5. My boyfriend currently is in school going into debt so can't contribute to retirement accounts... Would it be worth it to give him money for his IRA/401k?
Supermatthew is absolutely right - the boyfriend would need taxable compensation to contribute to an IRA. That said...you might not want to do that, even if you could.