To choose the correct size frame, measure the drawing. It should fit inside the opening of the mat. The mat not only looks nice, but it protects the drawing. The thickness of the mat, which is only the tiniest amount, is intended to keep the drawing from touching the glass. And yes, be sure the frame has glass. The only times you should use a frame without glass is with an oil painting or a poster.
I bought the frame you see here at Wal-Mart. There is no need to pay for custom framing. Frames and mats are generally sold together, as this one was. Once you know the size of the mat, the size of the frame will take care of itself. Look for a simple frame with a simple mat. I chose a mat with a nice outline on it to make the drawing "pop" a bit more. Whatever you do, don't get textures or decorations. A little bit of bevel is nice, but that's about it.
Before you slip the drawing in place, while the frame is apart, clean the inner surface of the glass throughly. Then assemble it all, be sure it's right-side up for the hanger, and then clean the outer surface. Hang at eye-level at a place where people can walk up to it for closer inspection. A fine drawing like this invites a closer look, so don't put it above the sofa or somewhere that people can't get close to it easily. I call this type of a drawing a "walk-up-to-it".
By the way, this drawing of a miniature dachshund was done by my brother Roger Hall and is of my wiener dog, Macintosh.