Agreed, gardening might be the best course of action at the present time.
Gardening is when you go a length of time without applying for new credit (no inquiries) and use that time to fix/remedy any baddies you have (accounts in collections, cards carrying balances, bankruptcies, etc.). This amount of time will also give your AAoA (average age of accounts) time to age. It might also be long enough for hard inquiries to fall off of your report, or just long enough for them to no longer affect your score. During this time, I would highly recommend seeing if any of your CCC (credit card companies) will consider converting your secured card to an actual credit card. I don't know if this conversion would require a new hard inquiry, so if it does then be prepared to weigh your options.
I would also recommend getting a CLI (credit line/limit increase) on as many accounts as you can, provided the lender doesn't require a HP (hard pull). Since it looks like you only have secured cards for the time being, this goal might have to wait a while.
I don't pretend to be an expert, but I would recommend gardening for 6 months to a year before applying for more credit.
EDIT: You ninja'd me.
Yes, Discover and Capital One give you your FICO credit score. Prequalified offers do not count as an inquiry, as those use a SP (soft pull).
EDIT 2: I just realized I might be able to provide some input on your original question.
Right now, it looks like you already have a Discover card, which is who I was going to recommend. The only other thing I would recommend from experience is to start banking with a credit union. I had a checking account with my credit union for about 6 months when I applied for my first ever credit card, and they gave me a $500 limit (which has since tripled). Credit Unions are much better than big banks because they are owned by their members.
If you have a family member that is in the military, I would highly recommend Navy Federal Credit Union (NFCU). They're known for giving respectable CL (credit limits).
