I would go to a store that sells a lot of basics, so any department store, and try on different things in different colors. Choose colors you like and then see if they look good on you. This is where a girl comes in handy. Bring a friend you know that will help you decide on colors.
My boyfriend likes to take me shopping to find out what looks good on him :P He always consults me, and I help him find stuff that he likes.
It's hard for anyone on here to tell you which colors, because it's highly subjective. Different people look better in different colors, because of skin tone, hair color, and even their personality (if that makes any sense). But once you figure out what colors you like, you can just start by introducing those things into your wardrobe.
Start with a t-shirt, and go from there. Maybe get some shorts with a design, or a sweater with stripes. Adding little bits of color can help introduce color without going crazy and feeling like you are changing too much.
Its hard to say without knowing what you look like. This definitely makes a big effect on it. Go for basic colors like white, grey, etc. Then try colors on and see which ones look the best on you. You want to wear clothes that compliment you.