I would allot 2 hours to drive from Gora to Kamakura, then another 1-1.5 hours to drive back to Tokyo (depending on where in Tokyo you're headed). Traffic can be unpredictable during the summer vacation season. While in Kamakura, you'll have to find places to park at each spot you visit.

Kamakura is a tourist destination for people visiting from Kanto, and is also home to a lot of commuters to Yokohama and Tokyo. As a result public transport is quite good (the Enoden, buses, monorail, etc.) and I don't know if it makes sense to rent a car if you're only planning to hit the highlights (Tsurugaoka Hachimangu, Hasedera, the Daibutsu, etc., which are the most well-linked by public transport). Lockers are available at Kamakura Station should you choose not to send your luggage ahead.

Unless you plan to visit places along the way that are inaccessible by public transport, I would stick with trains and buses. It will be cheaper and less stressful.