I spent a few years in South Korea, so I can give you a few answers:
1) best season to visit: NOW. The best season to visit Korea is Autumn (fall). October and November are terrific - the weather is cool and crisp, the trees are changing color, and the natives are happier. Second best is spring (April) when the flowers are cherry blossoms bloom and weather again is pretty good. Winter is cold and snowy, but creates beautiful landscapes and sets off the temples pretty well. Summer is sticky and humid. Avoid it.
2. Itinerary. Best thing to do is go to a travel agent that does group tours. You don't have to join the group tour, but you get some good ideas from it. Must see places include Kyongju (the ancient capital), Seoul (palace, temples), Busan (fish market). The rest are pretty similar, depending on your interests. Personally, I love the fish markets on the coast and the mountains. Another place of interest is a small town whose name I have forgotton which lives in the traditional way. It draws a lot of local tourists, but not many foreigners. Avoid the touristy one near Suwon though - it's nowhere near as good.
3. Public transport is really good in Korea. Go by train to most towns, and you will find a tourist information booth, or at least a tourist map in English. You will find that Korean maps are usually in English and Chinese charaters, so I you can read Chinese it will be a help. You will find accommodation in towns near places of interest and can do day trips to where you want to go. Korea is a tiny country, so distances aren't a big issue. And taxis are cheap.
Also, for trips around Seoul, check out the USO. This caters largely to US defense force personnel, but isn't limited in any way. Probably the best way to go to Panmunjom.
4. There is a significant communication problem and the Koreans try their best to overcome it. Me, I'm white, so there was never an expectation that I speak a word of Korean (even though I do). If you are Asian, you might not be spotted as a foreigner immediately, and will have to be a bit patient. Having said that, Koreans really do love showing off their country and talking about it, so it's surprisingly easy to get by. Target the younger people when asking directions or information because their English is way better than the previous generation. As far as electronic communications go (internet), you'll usually be within 100m of an internet cafe anywhere in any commercial district.
5. Suggested budget for hotels. You'll need to spend $30-$50 a night minimum at most places. Korea isn't exactly a backpackers' Mecca, so dirt cheap accommodation simply isn't available. On the other hand, for $30-50 a night you'll get a small clean room with ensuite, TV, aircon, heating etc. These small cheap hotels are available in any town and you can usually get a room any time. Make sure you avoid Chusuk and Chinese New year though, because the whole country mobilizes and transport and hotels are booked out.