How can you slim down your arms without bulking it up? Should I still do weights? I know women don't have enough testosterone to build up but I just want to prevent my arms from getting bigger? I want to look slim not big.
The only way you can slim a specific body part (non-surgically) is to do exercises that focus on that body part. And you can't do that without adding at least a little muscle. I'd recommend the standard: chin ups and push ups.
The biggest thing is to not use too much weight. Body builders do short sets of repetitions of heavy weights to add bulk to their muscle. To just tone up an area you need to do the opposite-use little weight but more repetitions. Yes it's tedious, but maybe you can work it into your daily routine, like doing some at your desk at work if no one can see you, and some when brushing your teeth (not sure I would be co-ordinated enough but still!).
You won't bulk up unless you really go for that by changing your diet and working out a LOT. If you just do some basic exercises (half push-ups where your knees are bent, or lift dumb-bells or bottles as someone else suggested) then you'll be fine.
Your muscles will get more toned, and better definition - you'll be able to see them better as the fat slims down, but you won't end up looking like a body builder