I second Revelate's card recommendations, with the addition of Barclay's (Apple or Cash Rewards are anecdotally among the easy-to-get ones).
I don't think it's unreasonable to have those goals. Look like you're well underway to getting your scores where they need to be. Small steps for sure. Getting those irritating sub-primes out of your wallet is something every creditbuilder wants to do. Without these forums I wouldn't even know what to do to get into Primeland. The Wally card will be a safe bet though. No AF and it's actually kinda useful, and I don't even shop at Wallyworld all that often and if I do it's just to buy ammo. I want that CF card myself, but since I just got my CSP I'm not willing to take another HP just yet for it. It's definately on my to-get list.
I think you can do better, the Credit One is really doing nothing for you and are you sure that their fees are waived for you? All the pre-approvals I keep getting from them are laughably ugly when it comes to their fee structure. I'd close it and not look back honestly.
I don't agree with cdnewman's analysis in this case (though I usually do FWIW), there are plenty of cards around the 640-660 mark which you appear to be at.
Amex entry charge card (Green)
Walmart (really it's below this anecdotally, if you shop there it's worthwhile I think, maybe for the FICO alone depending how you value that personally).
Get through closing, take a breather, and thwn when you come up for air take a look around and figure out what your spending pattern is going to be post mortgage (need home improvements in the future for example?) and then come up with the cards that make sense for you over the next year or so and pick one or two.
I haven't noticed their bullsh*t $7 something a month charge on it for the annual fee this year. I'm fairly certain that my husband was able to get the APR lowered (just for 6 months, but it's been at a 0 balance since then) and the AF waived for a year. Back in the land of the 4 and 500s, I didn't care too much about the $7 a month..I was just happy to have a card. Nowadays, though..that could be a nice hot starbucks coffee treat
Lowes would be a plus because I have a few things I'm itching to do to the house. It's turn key, so I don't have to do them right away, just somewhere down the line.
I think you need to be at the 680 plus range before you start thinking about getting Chase.
You should look at Walmart when you're at the 650 range.
Lowes, 650ish range.