Does the Audioengine A5 work well with a turntable? How good is the sound?
with a turntable unless it has a built in phono preamp or you connect a phono preamp between the turntable and the A5s. Audioengine A5s are good active loudspeakers. How good they will sound with a turntable may depend on the table, cartridge and phono preamp used.
What turntable and preamp do you recommend?
An entry level table like the Rega RP-1, ProJect Debut III or Music Hall MMF2.2LE will be $400-500. I haven't heard the ProJect Essential, but it is a budget table for $300.
An adequate budget phono preamp is the Technolink TC-760LC for $74.
What are the best turntables with a built in preamp to use with these speakers?
and the Technolink TC-760LC phono preamp for $74 before any turntable with a built in phono preamp. If you want good quality sound that would be the bottom. Maybe the Marantz TT42P at $349, but I doubt it is as good as the previous suggestion.
The Musical Fidelity VLPS phono stage preamp is excellent. It was $149, but is now selling for $99 at Audio Advisor, which also has turntables.
While you are looking at Audio Advisor, check out the Thorens TD-158 turntable for $299. That might be a good choice.