First of all, great idea! Make sure to get some basic knowledge of the culture and how things work here beforehand, though.
I guess you could pretty much see most of Estonia in a week. However, if you plan going camping/hiking or just relaxing, 10 days could fall a bit short. It all depends on how much you want to see, what interests you, etc.
You could probably look into using buses. The bus coverage is generally pretty good in Estonia and you can probably get almost anywhere you would like to by bus. The tickets are pretty cheap, too.
The overall cost of the trip shouldn't be anything outrageous. We have pretty low prices here for most of the things you will need to buy. The buses are cheap, train tickets are cheap, groceries and commodities are relatively cheap. Technology is quite expensive though, but I wouldn't guess you are coming here to buy computers anyway.
What to expect. Well, first, you should not expect good weather. It largely depends on the season, but we generally have pretty crappy weather. In winter, be prepared for under -20C (not this winter though. This winter sucks.) In summer, we generally have around 18 - 22C. Anything above 25C is very hot for us. At 30C, people have been seen melting into puddles on streets. All in all, you should be prepared for the worst.
You can probably expect to see a lot of nature and forests, especially when travelling in the countryside. We have really beautiful national parks (Lahemaa Rahvuspark, for example) which you should really visit if you have the time.