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, atypical for Japan, a car city with a very good road network and very sparse public transportation. Which is actually quite fitting, considering Toyota is a popular car brand!
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.
Toyota is well known for Toyota stadium which is the second biggest soccer(ball games) stadium in Japan. It has 4.5000 seats with 141m×88m ground and 115m×78m pitch side. International and national soccer and rugby games are held every year as well as music and community events. It is located right next to Yahagigawa, a 117km long river that crosses Aichi prefecture, Nagano prefecture and Gifu prefecture. Homi danchi is a unique town (multi-unit apartments) where a lot of foreigner live ( relatively people from South America such as Brazilian and Spanish people). It is a community where you can find diversity at schools and work places. There are super markets with many products and food from overseas unlike a lot of places in Japan.
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.