There's nothing to compare with seeing the night sky with your own eyes (given a clear night, suitable optics to help you see things better, and a human or other guide to help you understand what you're seeing). But thanks to the internet, there are other options that let you explore the night sky through your computer screen. Some amateurs such as Rob Gendler make magnificent use of digital night sky images captured with remote controlled telescopes. There are "pay per view" sites with elaborate interfaces, multiple large telescopes in good viewing locations, guided tours, and more (Slooh is probably the best known).
I found a free service that is not so elaborate but is an interesting place to start. It's the Micro Observatory Online Telescopes offered by the Harvard-Smithsonian Center for Astrophysics. Although this is an educational project aimed at teachers who will use the telescopes to enrich science education for grade 6-12 students, there is also an all-access "guest observer portal." The telescopes aren't very large, but they let you do some interesting things, though somewhat indirectly.
I tried this out the other day and received my first few images today. Basically this is telescope "batch processing" with a limited number of pre-defined observations that you can request, and a few "decisions" you have to make about the observations. For the Andromeda Galaxy shown above, I had to choose the exposure time (but it tells you if you choose the wrong one). For Jupiter and its moons (recreating Galileo's first telescope observations from January 1610), I just had to request a number of observations (up to 5 to be taken at different times, so you can see the moons' positions change). I made my requests, gave an email address, and two days later I received my images by email, with links to more detailed information online.
I was lucky and skies were clear last night. The images come back as GIF files but you can also download your images in FITS format (an astronomical image format that preserves more image data). To work with FITS, you can also download a free astronomical image processing application (written in Java). For the image above, I used this program to play around with the contrast, exported the result as GIF, opened in PaintShop Pro, increased the color depth (to allow more image manipulation), and played further with the contrast to bring out the arms of the Andromeda Galaxy. For some reason, I didn't receive all the Jupiter moon images I requested, so I didn't try out the option of making a stacked image and saving it as an animated GIF. I'll have to try again.
There's much more to explore here. While there are relatively few objects to grab, there's good background material to help you understand what you're seeing, and there are hands-on possibilities for learning to manipulate astronomical images. Rob Gendler, watch your back!