As Uji mentioned there are a few other threads but I thought I'd offer a few tips, as we went last year and had a great time (except for the rain!)
We used the normal JR trains to Suzuka Ino Circuit station. There was no big signage in Nagoya station for F1 fans or anything like that so we just made our way to the required platform and hopped on the next available train. Ask the station staff if you are unsure. At busy times you may have to skip the first one and wait for the next one. You pay the supplement fee at Suzuka Ino Circuit station when you arrive, they have two separate queues, those with normal tickets and those with JR Passes and the like that have to pay the supplement.
The walk from Suzuka Ino Circuit Station to the race track is not short (15-20 min), and is partly up hill. How long depends a bit on where you are picking your tickets up from/who you purchased from.
Once the day has finished and you walk back to Suzuka Ino Circuit Station there will be two queues, one going to Nagoya, the other one going to Tsu. In general the Nagoya one is the longer one but ask if you are unsure. When we left on the Saturday this was banked up for a couple of km outside the station and we queued for a bit over an hour to get on a train back to Nagoya.
There were a few of us in our group, and we arrived at different times and took different methods (some went Kintetsu trains and then by bus), but in general we all agreed that catching the train to Suzuka Ino Circuit and walking was the best method.
Book your F1 tickets, and accommodation as early as possible. Note that there are no General Admission passes, but you can buy the cheapest "grandstand" pass which tends to be a spot on the grass on the hill, and you can then walk around the circuit from there.
We went for a ride on the Ferris Wheel which was well worth it, buy your ticket for the ride from the vending machine at the base of the Wheel. Note that the Ferris Wheel is technically "outside" the circuit so you must have your ticket on you so you can re-enter the circuit precinct.
The souvenir shops are scattered around the circuit and there is one major one near the main entrance. Buy your souvenirs early as they may sell out of whatever you want.
Beer was priced at 500 yen for a 500ml can, food was reasonably priced too with five yakitori skewers for 500 yen, and six Takoyaki for 600yen. They also had beef yakitori, fried noodles, and a good variety of other food.
In order to avoid the queues for the trains, and because the food was so good, we actually stayed back after the race on Sunday and drank beer and ate food for a couple of hours before heading back to Nagoya. They also replay the whole race on the big screens in front of the main grandstand/pit area, so you can go and sit in the grandstand and watch the race again on the big screen if you like, even if you didn't have a ticket for that grandstand.
While you are in Nagoya, if you want to do the Toyota Factory Tour then book well in advance as I would guess the English tour books up a bit around the F1 race dates. It takes a bit of time to travel to Toyotashi to do the tour, but the japan-guide page and the Toyota Reservation page explain it all fairly well. If you don't have time to do the actual factory tour then the other Toyota Museums in Nagoya are both worth a visit.
If you don't want to deal with travelling back and forth to Nagoya then you can consider staying in Tsu, just bear in mind it is a smaller town with not as much to offer, but quite a few fans do stay there each year too.
Enjoy your trip!