1. -[te form
2. -[volitional form of verb -おう or -よう] + とする , on the other hand, emphasizes the fact that you are going to try, regardless of the result. Note that In fact, a similar construction (-ようとした), indicates failure ("I tried..., but I failed.).
As Elenkis said, there is a difference between “I’m going to try drinking sake (and see what what it’s like)” and “I’m going to try to finish my homework tonight”.
Other similar constructions are [plain non-past form]+ ようにする. Another is [plain non-past form] +ことにする.
3. [plain non-past form of verb]+ ようにする means that you put your mind on something, and you're going to exert all effort, even if you fail, until you reach your goal.
4. + ことにする means that you have decided on doing something. It can refer to a single or a habitual action. This seems close to the construction -[volitional form]+と決める（きめる）, as mentioned by Tae Kim, although I haven't heard or read the latter used.