You'll be doing a bunch of different exercises, so, unless you can do all those exercises using the same weight, you'll need a variety of weights.
Go with what you can do comfortably and also get the next weight up. You'll be getting stronger and able to handle it soon.
Personally, I purchased a set of exercise bands. I like free weights better, but the bands were a whole lot cheaper and covered the full range of weights I needed.
Good luck!

I have weights ranging from 3-20lbs (3/5/8/10/15/20). My husband used the larger ones but after 3 weeks of p90X I am already using the 15 and 20lbs on some of the leg and shoulder exercises. I rotate between bands and dumbells and have a tough workout using either of them. good pullup bar is a must, but with our living room there is no way to put one up so in the archway we put up a bicycle metal hook and I loop my band around that for pullup moves.

Hi, I had a set of older style dumbbells where you take off one weight put on another, etc. Shoulders and arms made me crazy because they move so fast from exercise to exercise. I couldn't change out the weights fast enough. I ended up buying an adjustable weight set from Sports Authority (SA Gear). It ran about $200 for the pair, but it was a lot less expensive than the similar Bowflex and Golds Gym models. Each dumbell has up to 50 pounds (plenty!). Just slide the dial to the correct weight and go. Love it!