How should you schedule applying for new credit cards if your goal is to not alarm prospective creditors while getting high initial credit limits? For example if approved for a new credit card should you apply for another before or after receiving and activating the new card / before or after the new card shows up on your credit report? I'm in the United States.
You should apply to all of them in a short period of time, rather than waiting.
Each time you apply for a card, it generates a hard pull on your credit record. These are negative events, but they are not all treated the same way. If there are a bunch of pulls in a short period of time (within a week), it is treated almost like a single pull. The ratings agency will assume you are shopping around for the best loan or card. Applying (and prequalifying) for a ton of mortgage loans is a completely reasonable course of action that doesn't indicate that you are a poor credit risk. The ratings algorithm supposedly takes this into account.
If you have a pattern of making applications over a period of time, on the other hand, their algorithm may interpret this as a person attempting to supplement their consumption with an ever expanding amount of credit, which they may think is a bad thing.
The actual algorithms ratings agencies use are proprietary, but still relatively well-understood. I obtained the above information from a loan officer, but I have heard it from several other sources and believe it to be more-or-less common knowledge among people who work in this area.
Note: This answer applies to the United States.
The answer probably varies depending on your current credit report and current income. In general, each of the applications will generate a hard pull and may slightly affect your credit, and each of the cards you accept, once they appear on your report will have a larger effect on your score, but that could be positive or negative depending on your current report. For example, typically the new account(s) will lower your AAoA (average age of accounts) which is a negative, but without a balance yet they should also lower your utilization percentage which is a positive.
Some people recommend that you apply for them all at once, so that none of them know about each other and they will be less worried about you spreading yourself too thin. If you space it out the later banks might lower the limit they extend to you, or possibly even disqualify you if they feel your current extended credit is too high compared to your income.
However, I personally don't like to recommend applying for everything at once, simply because the credit system works pretty well as it is, and if you have to play games to "fake it out", then that probably means you're trying to go into more debt than you can handle.
My vote would be to do whatever makes sense for your current financial situation, without worrying too much about your credit score, and without having to hide from potential lenders the fact that you already have (or will soon have) new credit with other lenders too.
This is also a united-states-based answer.
If you're going to apply for them all, and your existing total credit card utilization is low, apply for them all at once.
Once a new account is posted, the average age of your accounts drops, but so does your overall "utilization" of revolving credit. Both of these are significant factors in your credit score, and the effect on your credit score is negative from the first effect and positive from the second. Legitimate sites that provide you with free access to your credit score also often have simulators so that you can see how those factors balance against each other, and you can test to see if your credit score would go up or down with that additional account being added. They'll ask for a credit limit as an input parameter; you can run it multiple times to test across a wide range if you want.
In general, if your utilization is already quite low, lowering it more is not likely to help you much, while lowering average age would hurt the score more, and you'd choose to apply all at once. On the other hand, if you already have a relatively large number of old accounts and relatively high utilization, lowering your utilization (by waiting for the first account to appear) might be more helpful to your credit than the downtick caused by lower average account age. So the correct answer is it depends on your situation.
As of this posting, the current top answer focuses on inquiries and advises an all-at-once approach to minimize the effect of multiple inquiries. While multiple hard inquiries are sometimes grouped together to not punish people for "loan shopping" (which is good to encourage), that only applies in certain categories like auto, home, and maybe student loans, not credit cards. (See answers to this question from MyFICO (2), NerdWallet, or a HuffPost interview with folks from FICO & Experian.) If you've got inquiries that will be aging off the report soon, you could time things so that in each application there are only 1-2 inquiries on the report, but this has a relatively minor effect on your score. Also keep in mind that the longer you wait to apply, the later those inquiries will age off. In any case, this factor should probably not be driving your decision. Also keep in mind any given creditor might only pull one or two reports instead of all three, so the new applications might have even less of an effect.
If you're going for an intro offer with a requirement that "you must spend $X in the first Y months of the account to get a bonus of $Z," and you choose to pursue that because you would have spent $X on some credit card in that period anyway, great. If you've got two of those intro periods going at the same time, though, is your regular spending really greater than $2X, or would you get more of a bonus (without extra spending) if you staggered those start times? Think about those interactions when you consider the benefits of applying for each card, and only apply for a card if and when the benefits are "worth it."