Mounting downhill ski bindings is a simple job. The only challenge is steeling your nerves and drilling into an expensive brand new pair of skis! Ski shops often charge around $50 to mount ski bindings, a job that takes them only a few minutes. It will take you longer, but this is a straightforward job that anyone with a drill can do.
The hardest part about mounting skis is lining up the bindings on the skis. All skis have a centering line over the "sweet spot". This line matches up with the centering line on the middle of your ski boots. These are actual indication lines on the skis and boots. If you can measure the bindings so these lines match up, and don't mind drilling pilot holes in your skis, then you can mount bindings.
The heel part of the binding adjusts forward and backward (with a screw on the back of the binding) but the front does not. Accordingly, measure from the front backwards to make sure that the line on the ski will always line up with the line on the boots. If the boot doesn't fit perfectly, you can then adjust the heel piece as necessary. The boots I am mounting today are about 10 and 3/4th inches long with 5 and 3/8ths inches from toe/heel to the center mark. Be sure to measure backwards from the boot toe lug receptacle on the toe piece of the binding (not any other part of the binding) to the center mark, because this is measurement you took from the boot.

With these measurements in mind, I laid out the toe binding piece 5 and 3/8ths inches from the center line. For the heel binding piece, make sure that the adjustment on the binding is in the middle of its range as described above. When you measure from that setting you will be able to adjust the heal piece later if the boot doesn't fit exactly right.
Once the binding piece is at the proper distance from the center line, and centered side to side on the ski, mark the screw holes with a magic marker. Ski shops have templates for the bindings which saves them some time. Your bindings may have come with paper templates. But if you're careful, it's not hard to lay the bindings out on the ski using the binding itself (or the mounting plate if they are separate pieces). Measure twice, cut once takes on special meaning when you're mounting skis!

To install the heel part of the binding, tie back the brakes. This will make it much easier to mount the heel section. As I mentioned above, measure the heel piece from the heel piece when the binding is in the center of its adjustable range. However to install the binding, you may need to adjust the binding backward or forward to access the screw holes. You may also need to engage the binding (click it down) to access the holes. Just like the front, use a marker to mark where you'll drill your pilot holes.

Most skis require a drill bit that is 3.5 millimeters. Though to be honest, I just eyeballed the bit after looking at a screw. The top sheet of the ski will be hard to drill through, but once you get through, the core will be much softer. Therefore, mark the bit with a piece of tape so you do not get carried away and drill through to the bottom. Should you do so, you can always
patch the hole with P-tex, like you would for repairing a gouge after hitting a rock.
After you drill the hole, the top sheet may rise up slightly around the opening. This must be removed with a chisel or the binding will not sit flat on the ski.

Once you have the holes drilled, simply lay the binding over the holes and install the screws. These are strong screws so torque them down tight. Check whether there is a gap between the binding and the ski. It should be completely flush.
While all bindings are different (and hopefully come with instructions) many bindings have longer screws for the toe piece than for the heel piece. This is because the toe piece is often thicker than the heel piece. But because all bindings are different, don't be afraid to dry fit the screws to see how far they protrude under the baseplate of the binding. They should all protrude the same length.
A note about epoxy. Many people fill the pilot holes with two-part hobby epoxy before installing the binding screws. The thinking here is that this keeps the screw in place and keeps out moisture. I've heard that people swear by this and other people say it's unnecessary. I've never done this myself and I've never had a problem. Though I don't see how it could hurt!
Also note that most bindings come with the tension set to its lowest setting. This is because the spring inside will eventually loose its power if set at high tension for a long time, such as during shipment and while it is sitting in the store. Be sure to adjust the tension (measured as a DIN number) according to your skier type before you go skiing. This setting controls when the binding will release in a fall (and save you from injury!). Therefore, more aggressive skiers need a higher setting than more conservative skiers. Keep in mind that tension in the spring will eventually decline and so on older bindings, the DIN shown may not be the tension that the binding is actually applying. Along these lines, it's a good idea to set the DIN to its lowest setting after your last day of skiing for the year so the spring isn't compressed when the ski isn't being used. Even if you do so, the interior spring will eventually wear out and you'll need a new pair of bindings.
As with so many other things I write about, this is something that an expert may chastise you for trying yourself. But, like so many other things I write about, there is no magical specialized knowledge that a professionalized ski tuner acquired after years of study. People in trades tend to overestimate the difficulty of most jobs to ensure the demand for their services. Don't be afraid to tackle projects yourself. It's fun and it will save you some money!