Use a floor/shop or central vacuum with the hose attachment with bristles.
Do not use a vacuum with a rotating assembly.
Vacuuming will get chalk dust up as well as most anything else on the felt. Brushing alone will not get the felt clean.
As long as the felt is applied properly, you will not need to worry about the vacuum suction loosening the felt.
If there is pet hair on the felt, vacuuming likely won't get all of it. Use a pet fur / lint roller.
Additionally, a pool table cover is highly recommended to help keep your pool table's felt clean. You won't have to clean the felt as often if it stays covered when not in use. Playing on dirty felt will lessen the felt's lifespan.
Due to the excessive oils, we do not recommend using a furniture polish on most pool tables. It will leave a "wet" and streaky surface which will get on your hands when playing and then transfer to the felt.
A dry micro fiber or terry cloth towel is typically best to simply dust the pool table off.
We have found Method brand furniture cleaner to be a nice wood cleaner without an oily residue.
Again, a microfiber cloth works great. Glass cleaner or a multi surface cleaner that isn't too aggressive is suitable for tough spots.