The best place to start is DoC's monthly best bonus list!
https://www.doctorofcredit.com/best-bank-account-bonuses/ (Well, it's the 2nd best place to start after reading our DISchurners Bank Bonus 101 post

) Imho Chase $300 checking bonus is the best way to dip your toes in. If you have the cash to float grabbing the savings bonus from Chase is quite easy as well. Discover also has an easy savings bonus to grab. I'd keep an eye on the Cap1 $500 Money Market offer, people were able to cumulatively get the bonus by putting 50k in bits at a time (25k in, 25k out, 25k in totally 50k, etc...)
but they haven't posted the latest bonus in a bit (they batch process) so I'd watch for DPs of the bonus coming through recently FIRST if you'll need to do it in bits and pieces before jumping- but $500 is the best offer seen there- I was thrilled with my $250 back in the day on that one, ouch!