It was in fact named after the car company (and not the other way round).
Toyota is rather difficult to reach by public transport. Despite being close to the Tokaido-Shinkansen, it has no Shinkansen station, and it does not even have direct connections to nearby Shinkansen stations. The closest Shinkansen stations are Toyohashi, Mikawa-Anjo and Nagoya, with Nagoya having the fastest connections to Tokyo, Kyoto and Osaka. Toyota's main train station is called Toyota-shi, and approximately one hour from any of the Shinkansen stations, requiring changing trains at least once.
The most direct route to Toyota city is to take the JR Chuo Line from Nagoya city to Kozoji. Change to the Aichi Loop Line and take that to Toyota. Be warned that the Aichi Line only runs about once or twice an hour.
Toyota is, untypical for Japan, a car city with a very good road network and very sparse public transportation.
There is a really nice bridge that looks like it is constructed out of dinosaur bones. It makes for some interesting pictures and a nice stroll in the afternoon.
Rickey's (on the 1st floor of the multi-purpose Vits building  100 m south of Toyotashi station, open until 2AM) is the only Western pub in town, and as such usually well-frequented by locals and Toyota expats from all over the world.