The capital of Georgia is often both an economical and fun way to travel. It is by far the least convenient. You should plan a few extra days for your trip, one coming to Batumi, and one returning to Tbilisi. However, you can normally find very cheap flights, excellent attractions, and affordable accomodations.
To get from Tbilisi to Batumi, we HIGHLY recommend you take one of the many daily trains, or one of the busses that go every 3 hours. The train is comfortable, smooth, and peaceful. There is even wifi included in first-class. First class tickets should run around 30 Lari ($12 USD). There are several daily trains to Batumi. With the recent completion of a modern highway from Tbilisi to Batumi, it is only 5 hours to travel by bus. Many bus companies offer comfortable buses. Some even have wifi. A bus should also cost about 30 Lari ($12 USD).