First of all... and this is probably so obvious you will already know it...
Not all web developers are alike, the old maxim 'You get what you pay for' is pretty much true, if your developers quote seems like it's cheap, it probably is - simply because they simply don't care about building a working relationship with you.
They will nod and agree without trying to educate you on what works and what doesn't, getting to what you really need from your site, because sometimes (and don't take offence), what you think you need isn't what you really need. They just want your money (we do too, but we also want you to be happy with the end result!)
We don't just take on any project, we have to know that if we take on your job that it's going to be done right, that you are going to get what you expected, and don't forget, if your developers language skills are 'iffy', thats going to follow through on your website.
Bad spelling, terrible grammar, it all adds up!
Next question is...
What platform are you going to use? There are other CMS systems out there, the ones to avoid are the 'bespoke' ones, these are created in-house by the developer, not a bad thing in some ways as they should know it inside out.
However... and this is a biggy...
If your relationship with them fails for whatever reason you are tied to a system that ONLY they can work with, in other words they have you over a barrel, it's stick with them or start from scratch.
ALL our sites are built using the Open Source Joomla! CMS, which basically means if we all get hit by a bus, or die from a badly disinfected keyboard (hopefully not tho') you have thousands of Joomla developers worldwide that can pick up the reins and carry on.
Ask for references..., seems obvious but you would be surprised how many clients don't.
Many developers show sites in their portfolio that are no longer operational or have been re-developed simply because the client just couldn't work with them anymore.
Contact their clients and ask them 'what's it like working with XYZ Co., any problems?' We think you will be surprised.