Day is your best resource when it comes to information on learning things as a player. More specifically, watch his "Newbie Tuesday" videos. On Tuesdays he specifically focuses on tips and tricks for new players to get started, how to get yourself prepared mentally, and how to have fun doing it.
To get over the "fear" of playing 1v1 you just need to go in with the attitude of, "it's okay if I lose". Because otherwise the anxiety will never have you playing solo games. I used to suffer from that a lot, even today I still have times where I feel that anxiety but nowhere near as bad as before. As you play more games, your attitude will change immensely.
To get better you simply need to play more games which goes hand in hand with the previous paragraph. But it's not simply playing those games, it's learning from your losses. Was your macro slipping (did you have too many resources and not enough production)? Did you scout enough? Did you recognize what type of build your opponent was going when you were scouting? These are all very important things to keep in mind when playing and analyzing your losses. Don't analyze your wins too much because you learn much more from your losses than from your wins.
There is a "multi-tasking trainer" on Battle.net which will help you learn how to micro, macro and spread creep. This map is good practice outside of regular games.
You didn't mention your race so I can't really give you advice there, but the general advice when it comes to races is to pick one and get good with one first. Then learn the others.
After you have picked a race, learn the ins/outs of every unit. What are your units good for? What are they good against? Also learn little things like what armor type are they? Do they have bonuses versus other types of units?
As far as watching SC2, if you want to see top level play, subscribe to the GSL. The top players in the world compete in this tournament year round. It's where the best of the best go to play and not-surprisingly, dominated by mostly Koreans. But you can learn a lot from listening to Tasteless and Artosis cast these games and see things that the pros use and try to emulate their play. (Although really the best I've done is emulate the strategy and it still works). Other tournaments such as the NASL, MLG, and DreamHack are really fun to watch as well.
The TeamLiquid and SCReddit communities are also pretty good resources when you want to keep up with what's happening in the SC2 world, ask questions about strategy, and looking for teams to play with/get onto one.
Any other follow up questions, feel free to post as a comment and I'll answer as best I can.