Welcome to the group!
Great question also great idea to exercise. Exercise will really help you to sleep and it will help your mood as well. When we are not getting enough sleep and we are depressed it can be very challenging to exercise.
Start small and try to do activities you enjoy. A gentle walk or a couple minute dance around the house can add up if done consistently. Don't put too much pressure on yourself. Even a couple of minutes here and there can help. If you can find activities you enjoy it will be more likely that you stick to it.
If you enjoy doing traditional exercising I highly recommend strength training, yoga and H.I.I.T. Strength training can be done at your pace, start with small weights and work your way up. If done consistently even for just 15 minutes, you will be surprised by the results. Weighted squats (or not weighted) are a powerhouse of an easy exercise. Look up a few strength training exercises online and then build a simple routine you will stick to. You can add more repetitions, sets or moves if you feel up to it bit keep the routine simple at first to ensure you stick with it.
Yoga is extremely helpful as an exercise and a stress buster. Look up an easy 5 to 15 minute routine online (youtube has tons of options) and see if you would enjoy adding some yoga moves to your exercise plan.
Lastly and most importantly H.I.I.T., high intensity intervals training. It sounds more intense than it has to be but it is very effective in a short amount of time. Simply do a cardio exercise intensily or a short amount of time and then decrease the intensity. For example, walk very fast for a minute or more and then walk slowly. I use a recumbent exercise bike and I increase my speed and resistance for two minutes and then go at a more comfortable pace for three minutes. I repeat this routine for an hour but even 15 minutes is an awesome start. You will be amazed by how quickly your stamina and overall health improves.
As a side note, if you feel your fatigue is extreme and your stamina doesn't improve with practice you may want to talk to your doctor. Fatigue can be caused by many health issues that may need to be ruled out.
What exercises have you enjoyed doing in the past?
Thanks for posting!