Advice for working out safely at home:
Hire a professional trainer
If you are planning to train at home with your own equipment, then I strongly recommend you hire a professional trainer for a couple of sessions just to get you started. This will not only avoid any injury as you are shown how to perform the exercises with correct form, but if you do the exercises with good, strong technique then you are more likely to get the best effective results and know exactly what to do in order to achieve your fitness goals.
Build up the intensity of exercise over a period of time
If you're fairly unfit, build up the intensity and frequency of exercise over a period of time. Do not go mad in the first few weeks, try and pace yourself and you will be surprised at how quick you can go from a beginners level into intermediate. Think about your safety when first starting out and make sure that you enjoy the activity you are participating in. Consult your GP if you are over 50, very overweight for your size or have never exercised before. I also advise if you have a family history of medical conditions or have one yourself which would potentially affect you if you were to exercise, it may be wise to have a medical clearance before exercising.
The importance of stretching
I recommend stretching just after a warm-up (static) and and at the end of the work-out (dynamic). You will achieve best results if you hold the stretches for a lot longer at the end of your work-out as warmer muscles are much more receptive to stretching then. This will also help to promote your flexibility and delay DOMS (delayed on-set muscle soreness). It is also fine to stretch during your sessions. The key is to listen to your body and work with it's guidance.
If your using equipment, I advise you start slowly and build up your duration or repetitions. If you are doing a cardio exercise, then build up the length of how long you keep going for each time. You should be able to speak in short sentences to know whether you are exercising to an effective enough level. If you can waffle on, then you need to notch up the intensity, however if you cannot speak at all, then bring it down a level. If you are doing strength bearing exercises, build up your sets and repetitions. To know that you are performing in order to get results you should be doing resistant work almost to the point of failure (until you physically cannot do any more).
Tips for stomach exercises
Ab and core exercises are good enough using your own bodies resistant or and using a swissball. When doing your sit-ups, the most important advice that I can give is to always keep your stomach muscles engaged. Make sure that you do not arch your back. Keep the spine neutral at all times so the back is always fully supported.
One of the most important things reminders I would like to point out with exercise is to BREATHE. It is so highly recommended! Always exhale on the effort (the powerful part). Most people forget to breathe when they are exercising as they are either concentrating on getting the exercise right or they are trying to get it over and done with.
If your motor-coordination is not great at first and you simply cannot remember when to exhale/inhale, then just breathe naturally. It's so important for your body to be fully oxygenated during activity. Remember 80 percent of your energy derives simply from breathing. If you enjoy what you're doing, then you don't have to think about getting it over and done with. If you are not a very motivated person when it comes to exercise, do a team sport or train with a partner so you both have something to look forward to and cheer lead one another along if one of you is not feeling energetic.
Have fun with it and be safe!