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Michael, just continue with what you have for some time until you have a better feel for observing. The Moon will be turning up soon after sunset in the next few days. You should be able to find that easily and that is exciting to observe. Your telescope will do a good job. If you want to get up well before sunrise!, Saturn is getting higher each day in the eastern sky. That will also be a good view. And, as I said before, Jupiter in the evening will be great!
It's not a question of how far away something is, it is a question of how bright it is. So, ultimately it is the aperture (diameter of the light-gathering part of the scope - a lens at the front in your case, or a mirror at the back for reflecting telescopes) that determines whether you can see anything at the eyepiece. Maybe better eyepieces will be a litle bit useful, but I'm quite sure that your 20mm eyepiece will be just fine. The 10mm will be a bit worse.
Your scope should also be useful during the day on terrestrial objects - it has what are called erect image optics in the diagonal (the part your eyepiece goes into). You can also learn a lot more about using your telescope at daytime when you ca see what you are doing. BUT - don't point it at the Sun. Nasty things happen to your eye and to the telescope.
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