Quality microphones do make a lot of difference. Having a dedicated room that you can adjust/alter for audio-recording (fully carpetted floor already prevents quite some echoes from voices in this room. But you can do more to the side walls and roof to eliminate the rest. As a bonus, this prevents noises from the outside coming into this room.
For each of those microphones, you will need a solid stand to hang them from. Preferably these stands allow the microphone(s) to "travel", so you can adjust accordingly when setting up a game with elaborate pieces. That, or a person keeping a microphone close to you, outside the view of the camera. By all means, if you plan to use more than one microphone, keep the recordings each on a separate audio track.
Get one or more decent cameras. It would be helpful if you can hang and stabilize these from those microphone stands, if you want have different angles while setting up/playing games.
The youtube show 'TableTop', part of the 'Geek and Sundry' channel, does a marvelous job with audio and video, angles and lighting. This show is done by a professional crew and it shows.
Personally, I think it is best to adjust the lights in such a manner that the camera(s) just need to record and not apply any compensation/adjustment options that are baked into their software. Diffuse light is your friend. Any editing you need to do is done best in software such as LightWorks.
Not sure if your use-case allows you to use LightWorks for free. It isn't the easiest software to work with, but it is used in a lot of Hollywood productions, so it delivers after you get the hang of it. By keeping all audio and video feeds on separate tracks, you can mix-n-match the feeds to make the best quality video-presentation you can...with regards to production value.
