Material: J A Milton or Relics in Witney (both webbable) Blues, dark greys, dark reds or black are other possible colours rather than green. Cotton moleskins also work well rather than baize - they're no good for billiards, but they're more hard-wearing generally.
As it's not for billiards, then it's easy. You can use any material, use any glue, and it doesn't even need to be ironed afterwards.
Useful glues would be Evo-stik 528, PVA (cold), PVA (hot melt) or animal glues. The animal glue you want is rabbit-skin based (or a secret mixture) rather than woodworker's hide glue. This is similar, but more flexible, and always a better choice for fabrics, leather or paper.
Evo-stik is the commercial solution, because it's quickest. However it needs accuracy and practice, because once it's down, it's _down_. I wouldn't use it. It's also hard to clean off in future years.
Personally I'd use rabbit on a fine piece, hot melt PVA on average work. Rabbit goes on hot and is applied "wet", PVA is applied cold and allowed to dry, then used as a heat-activatable contact glue by ironing the fabric down onto it.
Don't use wet PVA directly onto the fabric unless you've already tried it - you don't know how random fabric is going to react to moisture.
You'll want a couple of broomsticks pinned across the width of the fabric, possibly with some weights on, to stretch it taut when gluing down. An assistant to lift it on helps too (not so tricky for a small table)
Don't try to "stretch" it into place and not glue it at all. This can work, but it's enormously dependent on the quality of your fabric. It also needs ongoing maintenance and isn't necessary except for high-end billiards.
As always, experiment with your fabric, glue and a chunk of plywood before diving in for real.