There's no specific amount the limit starts at initially. I checked mine for $4000 the day I activated my card and got a yes, before running a single charge through it.
Others have seen their internal limit start lower than that. Others have seen higher starting limits. It all depends.
AmEx seems to have a formula that sets it based on your income, reported assets, credit history, and who knows what else. The number can differ from one borrower to the next.
I will say that since $4000 is my likely max monthly spend (barring an unusually large purchase), I did not "ease into" using the card at all. I put the vast majority of my regular spending through it from day one, and have seen no AA.
In your case, I would use the card as much as I can, paying in full when necessary to reset the balance. I would just avoid making lots of little payments. Use it normally, make a big payment, lather rinse repeat.
Starting: EQ 622 (myFICO 7/7/12), EX 696 (TU FAKO 8/14/12), TU 621 (CK TransRisk 7/24/12), Total CL $1k on 2 TLs
Current: EQ 709 (CCT 2/4/15), EX 704 (CCT 2/4/15) , TU 702 (CCT 2/4/15), Total CL $110.3k on 14 TLs Goal: 740+ x3
My Wallet: Amex BCP $30k, Chase United Explorer $16k, Amex SPG $13.5k, Barclaycard Ring MC $12.5k, Chase CSP VS $12.2k, Discover it $10.5k, C1 Venture VS $6.5k, Chase Slate $3.5k, Amex Hilton Surpass $2k, Barclaycard Apple V $2k, Chase Freedom V $1100, BoA Cash Rewards V $500, Citi BestBuy $500
My Loans: Prosper $25k/36mo, Prosper $17k/36mo
My Business: Chase Ink VS $5k, Amex BRG NPSL (> 10k),
