I want to take advantage of the lower rates... and I want to shop around at different banks (unless you think that is unwise).
What must I know/do before each bank runs my credit report?
First make sure you know the process for paying off your current mortgage in the refinance and make sure that refinancing makes sense for you.
Per the comments on my answer here, it appears that there is conflicting information about how long you can space out your shopping before the pulls start coming in separately. The shortest number quoted, and thus the safest, is within 2 days.
If you research your banks in advance and do the pre-approval process at all banks within 2 days, the pulls will count as one hard pull on your credit instead of a couple/few/several. In that case the ding should be about as minimal as it can be on your credit.
Do your shopping around first, pick the number of banks you can visit in 2 days to actually go through the process and get your good faith estimates from those lenders. Visit them in those 2 days. This will ensure a minimal hit to your credit.
Also make sure not to be doing anything else involving your credit, especially opening up new lines of credit, before you close.
EDIT: This link states that the window for FICO is 14 days.
Regardless of the impact on credit score, you should be shopping around for other factors before you start applying for loans or prequals. If you have good credit, you already know the rate -- the bank has it posted in the window. If you don't have good credit, tell the loan officer your score.
Closing costs can often make or break a refinance more than rate. If you live in a state like New York with an onerous mortgage tax, this is even more of a factor. Other factors may include mandatory escrow, PMI, etc.
Think of the mortgage process like the college admissions process -- you can apply to 50 schools... but 2 or 3 probably make more sense.