I just went a week ago to take a friend for the first time. There is not a lot to do wrong if you go, except maybe betting a huge sum. Just have a look at the race calendar to make sure an actual race is taking place, as often all that is going on are "satellite races", where the show races from other velodromes in Japan live for people to bet on.
For the foreseeable future there won't be any highest level races, but that doesn't mean it's not exciting to go to a lower ranked FII or so race. Most of the events span three days, with heats of riders competing against each other and the stronger riders usually in the later heats - leaving early is not encouraged. The latest day will see the most heated races and draw bigger audiences, as the final races made up only of qualified riders take place. So Sunday usually is a good day to go.
Entrance fee is just 50Yen and if you get a reentryticket (sainyuujouken) from the guard at the guards, you can go in and out as much as you want to. There is a conbini on the NW side and you can bring in whatever you want to as long as you don't go shirtless. If you want to, you could also get better seats inside with AC, but I always end up in the second curve, which is without extra charge and is roofed.
Straight after the entrance is an information counter, where some English is spoken. They can set you up with a ton of multi-language information material about Keirin, betting and the current events. To get a real wealth of information for betting, there are newspaper vendors, but of course those are only Japanese. Betting is fully automatized through vending machines. You write your bet on a multitiple choice style paper slip, throw your money into the machine (minimum bet is 100Yen), feed it the slip and out comes your receipt. Tell me what you do with the receipt once you win, I only had close calls so far
To get back to the core of your question - of course it is best to go to the Million Night Race at New Years. That is the highest level event in Keirin, when the champion of champions from that race year is crowned.