Do your research: look up dive centres on the internet
Once you've established you would like to do a dive course, start looking up different locations and dive centres that may be a good fit for the particular course you want to do or your specific interests. You may want to discuss with diver friends, dive pros you know, dive centres in a different location that you know or you may do what most people do: ask Google.
If you are doing your Open water course you might want to check that there is a good pool or protected area to start in and some exciting stuff to see, if you are doing your deep diver speciality or divemaster course, you may be more interested in what you might find at 40 metres, like Mola mola or getting experience in strong current dives.
Make a selection of dive schools you like
Once you've established where you want to go, use Google to list the best dive schools in that area. You might want to have a look at their website, see what PADI rating (or other agencies' rating) they have, and have a look at the pictures of their facilities and the general vibe of the place. Make a list and start emailing the good ones. The speed of response and the quality of first contact often eliminates some operations from your list immediately.
Get in contact & ask questions about diving and dive training
When you email the dive centre ask them all, I mean all, the questions you may have. They should not have a problem answering them in depth and even offering to chat with you on the phone if some information is still unclear. The openness to questions is also an indicator of the quality and consistency of their instruction.
Meet the people: dive instructors, dive professionals, dive staff
Though it is not common for dive centres to immediately assign an instructor to a given course, if you want to know who will teach you, feel free to ask, it should not be a problem to find out. Meeting owners, managers, staff and several of the dive instructors beforehand if that is available to you will already give you a pretty good idea of the vibe and professionalism of the dive school. Again ask questions to your heart's content, answering them should not be a problem for the dive school if they value their high standards.
Check the premises/ facilities of the dive centre
Why not check out the facilities like the pool, equipment room, compressor set-up, and oxygen unit/ emergency equipment beforehand? Most serious dive centres will be happy to show off their facilities to you and give you a full tour of their premises.
Make sure it's going to be fun: diving is fun!
Last but not least, check if the vibe and atmosphere are relaxed and playful, After all, you learn so much better when you are having fun!