When you add weight depends on diet a lot too. If you're losing weight, it's very unlikely that you will be getting stronger, althought is possible in certain cases. Even gaining weight, if you don't have the proper nutrition, weight gains may not happen all that often.
I'd say if your diet is perfect, either maintaining or gaining weight, and you don't see any improvements in an exercise for 2 weeks, then something is wrong. This is when I'd re-evaluate what exercises you're doing for that particular body part. Cowpimp also gave good advice as to when to add weight.
Also, after a couple months, you may stop seeing gains altogether. If this happens, try a completely new weight training program! Try to do a little research at a time while you're doing your current workout, and by the time you're sick of it, you'll have something new to try. There are a million different weight training techniques.
If you're new to lifting, start slow! After a couple weeks you'll start to see what areas need more or less work. If you don't know where to start, ask people here at IronMagazine to put together a beginner plan for you.
As far as staying safe, there are a few things you could do. If you have a friend who knows what they're doing, ask them to come to the gym and help you out. If not, it might be work investing in one or 2 personal training sessions to help you with your form. You can probably make another $50 bucks one day, but you won't get another back. The trainer can also help you put together a workout plan, although they will probably try to get you to buy more training sessions too
Listen to your body, and record your results. Don't forget to eat well. Doing all these things take some work to get into the habit of, but are second nature once you've been doing them for a while. Keeping a journal on this website can be great too, as a lot of people here can really help you with your plan.