Congratulations on your new Wii and welcome to Wiichat!
The skill levels in Wii Sports don't serve much purpose, other than to show you that you are getting better at the game and give you something to aim for. They only increase in single player mode (except for bowling and golf).
You'll probably notice that in some sports the levels will change quite a lot (especially bowling) but in some others (boxing, tennis) it will take a long time for there to be much change.
When you reach a certain skill level in a sport you get 'pro' status, which alters the Mii's appearance in some (bowling ball with stars on it, different coloured boxing gloves etc).
If you enjoy Wii Sports and would like more of the same then don't buy any of the knock offs (Deca Sports, Carnival Games, Game Party etc). Wait for Wii Sports Resort to be released later this month which increases the amount of events to 12 (only 5 in Wii Sports) and adds in the motion plus attachment for improved control.