Accommodation around Sun Moon Lake tends to be expensive, but if you can stay during a weekday you should be able to negotiate a cheaper rate. A day trip by bus from Taichung is possible, but unless you're driving a rental car or otherwise have your own form of transportation, it's better to spend a night there. Generally speaking, accommodation is less expensive in Idashao than in Shuishe (aka Sun Moon Lake Village, where the tourist information center and main bus stops are located), but Idashao is harder to reach if you're using public transport.
In many respects, Sun Moon Lake is arguably the most overrated attraction in Taiwan, so you shouldn't feel that you have to make a special effort to get there. If you're determined to go, and your itinerary allows for it, I'd suggest approaching Sun Moon Lake via the scenic Jiji Railway Line. Bicycles for exploring the countryside can be rented at a couple of stations along the line (Ershui and Jiji), and there are frequent buses running between Shuili Station and the Sun Moon Lake visitor information center in Shuishe Village.