I absolutely agree with the increasing the CL with your Cap 1 Secured card. What a great idea!
IMHO, however, I don't necessarily agree with the continuing to app for anything for a while though I do see merrits to the argument to do so.
Just a word of advice regarding Barclays... they are very app sensitive so if you app and get approved, don't app for at 6 months afterwards or you really do risk getting shut down with your limited credit experience. But as always... YMMV and I wish you the best of luck. I really think you are off to a GREAT start!
What are the requirements for the Amex Zync? I've been lurking here and I'm in a similar boat as the OP
I think minimum requirement is 6 months credit history which could generate fico score. min ex fico I saw people get approved is 620 but that's rare. 650 would be safer.
Not too many inquiries, preferrably less than 4.