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Congrats on starting the process. I was right there with you about 15 months ago. I started with the Quicksilver and the Quicksilver One card with Capital one. After having those two, you can move over to Barclay and apply for the Barclay Ring and the Apple Rewards Card. Make sure your utilization is low when you do apply. I got started with those four and am rolling full steam ahead now. Good luck!
Both my wife and I applied for the QS One and Classic, both were approved for $500 for each of us, after five or six months the limits were increased automatically to $2500 each for me andlike $750 each for the wife - never could figure out why the difference.
After the increase we also both applied for Barclay and were both approved for $1300, where they have sat for over a year...
That being said, I'd apply for the Barclay card first then the Cap 1 QS, those three will optimize your score. If you belong to a credit union speak with them about their requirements, NFCU gave me a card very quickly and they have been very generous in my CLIs.