Go to inventory and right click on the item and select wear. It should appear on your avatar.
Because the clothing items and textures are stored on different servers, it can take a while. There are some things you can do to help speed it along.
Press CTRL, ALT and R at the same time. This rebakes the textures. They will go blurry for a few moments and then settle more clearly. If it's still not having it, try right clicking on your avatar and select the Change Appearance option. You don't have to change anything, just wait for the Appearance window to appear and then exit out.
If after all this, you still don't see your clothes, it could be that the server has corrupted them. To find out if that's the case, go to one of the places where you picked up a free outfit and get the same thing again. Try wearing it and see if it's better than your previous outfit. I had this with a free outfit from Nicky Ree. I just went and got another copy and that worked fine.
Just for your information, if you buy an outfit and have a problem like this, contact the creator. Right click on the item in your inventory and select 'Preferences' and then click on the 'Creator Profile' and choose IM and explain the situation. Very few creators refuse to replace an item that has become faulty - even if it's due to something that is not their fault. Just remember that if you don't get a reply within a few days, try again, as most busy SL creators get their IMs capped - I get IMs capped all the time, so if you buy anything from Rivermoon (my clothing brand) and have a problem, just keep IMing me until I reply. ;)
Here's hoping some of this helps.
Over a year in SL. Helping lots of new players. Clothing designer in SL.