For designing bridges in the USA, the AASHTO Bridge Design Specifications would be the one required reference. Beyond that, it depends on what type of bridges you're going to be designing. However, learning bridge design on your own would be a arduous task. I hope you have an experienced bridge designer to mentor you.

Anyway, I can't recommend any books on the subject from personal experience, since I haven't actually read any - I just learned on the job, applying what I learned in school to the loading conditions that were given, asked a lot of questions of the experienced bridge design engineers in the office, and carefully studied their designs. The bulk of my 'book learning' has been from NHI courses, which is probably the only way to get up to speed on bridge design in a short period of time. They're expensive, though.

For books on the subject, all I can suggest is to search on Amazon and look at the highly-reviewed results. Hopefully, others can help you with more specific answers on that.

(From UK Perspective).

I'd say 50/50 its learning from others and from the SCI guides for steel-composite bridges. The Designers Guides to Eurocodes are great as well, theyre not just to discuss the codes but also contain a lot of practical bridge advice.

I owe a lot of my early career success as a bridge engineer to nerding out over these on the train in my first couple of years. Read read read read.

For design in the USA, I recommend "Design of Modern Highway Bridges" by Taly. Very easy to follow, many different bridge types, lots of examples, very thorough