I would recommend contributing to a Roth IRA until maxing it out. These can be opened up any number of places depending on your specific needs. Check out vanguard, e-trade, share builder, tdameritrade, schwab, betterment, etc. Stay away from flesh and blood brokers. I have no idea what your bank would offer.
Vanguard - is probably the best option for you. Their mutual funds and ETF's are low fee and ETF's are transaction fee free as well. Stock trades are $10 or $7 depending on how much you have invested.
Since you will have a small starting amount and will be making frequent, small transactions for the near future - I would recommend just picking 2 or 3 of their ETF's and alternating which you invest in each month. This should give you the lowest possible total fee, dollar cost averaging, diversified investment you can get. I would just set this up on auto-pilot as much as possible for the time being - you just won't have enough to invest for a while for it to be worth your time. Focus on finishing up paying off your debt, improving your income, and improving your DIY skills.