Putting printed pictures in the right photo frame size?

As is, the photo has white space around it.  So if you place it into an 8.5" x 11" frame, you'll have that white space showing which doesn't look good at all.  That white space can easily be cut (best to do it with a mat cutter) so that it fits inside a smaller frame. 

Often a matboard will be used.  A hole will be cut within the board and the photo, which is much smaller than the board, will be mounted on another piece of matboard  which is then placed the behind the matboard with the hole (window).  This yields a much more professional look.  Your local arts & crafts store or anyplace that sells frames can do this for you.If you look, you're likely to many 8.5x11 frames as you are to find 8" x 10" frames.  It is your choice as to what you want to do.

Aesthetics, personal preferences, budget. It's all up to you, what you think looks better, how you feel about it and what fits your pockets.

You can use a 10x15ft. frame to drive a mocking point. You can use a slightly larger frame to create the illusion of a second frame. You can use a smaller frame to create a narrower perspective. You can use a frame that fits exactly because that's what most people do. You can use a circular frame because you're outgoing or a square frame because you're square. Best of all, you can use that old frame in the attic of whoever lived in your house previously so you won't have to pay a single cent. There is no rule. You can even just tape the picture on the wall, frameless and all!

Two ways of doing this.

You can get a frame for 8x10" pictures. This is a very common photo frame. These frames generally chop off a bit at the edges. This means you need to trim your paper. These frames are the cheapest way to frame a photo.

You can get a larger frame than 8x10 and mat your picture. This generally looks better than the previous. There are  frames that have precut mat for 8x10 images and easy to source at your local art shop or on-line. Or, you can custom cut your mat. If you get this done professionally, it will become expensive. Cutting your own mat is fairly easy, but you need a mat cutter which runs about USD100. Acid-free mat board is pretty pricey for what it is.

Use photo corners to mount your picture to the backing. Don't mount your pictures with tape or you will get wrinkling when moisture levels change.

I'll offer a third way which is mounting the image to foam-core. This doesn't use a frame and is cheaper than framing. For certain images, this isn't too bad. Use photo-mount spray. I use this technique for photos I hang in my office.