No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
I'd wait until they are 6 months old, unless you want to try for a store card.
What time next year are you planning to purhase? If we're talking first half, I'd just sit absolutely quietly on my existing tradelines and let time work as much magic as possible in terms of score raising. If we're talking back half of next year, YMMV but you likely aren't qualified for the good entry-level unsecured cards (Chase Freedom et al.) yet, so I'd probably still hang out and finish out the mortgage process... which is actualy important vs. credit card chasing which is comparitively trivial.
In fact that's what I'm doing right now .
I would wait until about 6 months.. unless you call the credit company and see if they can/able to convert you to unsecured... no harm in trying...