I am wondering if people usually fill a raised bed completely with soil, or make a "bottom" for the soil to sit on? If a bottom is made, is it treated wood to avoid mold and deterioration? And if drainage is necessary, how do I create that?
What people "usually do" depends on specifics of local economics, climate, and availability of materials.
I have a friend who lives in the Denver area. The soil there is awful and buying topsoil is very expensive. He spent $2,000 on lumber to build a backyard full of raised gardens with benches, then spent another $1,000 for different gravels to fill most of the boxes. Then there was $2,500 worth of topsoil brought in. The mesh canopies were comparatively a bargain to protect all the produce from being smashed by the daily hail storms.
Here, in Oregon's Willamette Valley, we have great topsoil in most places, so here we usually just shovel it some from somewhere we don't need it to fill the planters. Or if there is some otherwise unneeded concrete rubble or gravel, that would go loosely in the bottom to decrease the amount of soil needed to fill. Every year new compost goes in, and those who don't test their planter soil seem to buy bags of amendments too.
If the raised bed is sitting on the ground no bottom is required. A bottom would only be required if you are protecting the surface below the raised bed. If the raised bed is set on a flat roof, a wooden deck, patio that would be subject to staining by the soil, etc. then you would want a bottom on it. You still need to allow for some drainage. A word of caution about using treated wood. Most treated lumber contain toxic chemicals that can leech in the soil. There is the potential that the plants can absorb the chemicals. Not a problem for a flower bed but a concern if you are growing veggies or other edibles.
The main purpose of building raised beds is to be able to grow in a custom soil mix because your current soil is poor. The raised bed soil is usually lighter, fluffier, holds more water and has more nutrients. It's more of a potting soil.
Most flowers don't have very deep roots, especially annuals, and if you have a big difference in soil textures between your existing soil (hard) and raised bed soil (soft) the roots will grow thicker and more dense in the potting mix and become thinner in the existing soil. Look into which plants you plan on growing and see how well they do in soil as deep as your raised beds.
You don't need to put a hard bottom in your raised beds but many people use a weed fabric to keep weed seeds from germinating in the existing soil. Weeds can dig through quite a bit of soil to reach light if the soil in the raised bed is soft and light.
Weed fabric will block weeds while still allowing drainage.
If you're putting raised bed right up against your house (or other structure) you may want to put some water impermeable fabric on the bottom as a slope to direct drainage away from the house. Dig a small bit of the soil under where the raised bed is and slope it away from the house. Line it with 6 mil poly and leave it open in the front for water to drain from there.
not necessarily talking about the same issues, but worth mentioning: A .5" x .5" galvanized hardware fabric would be a good "bottom" of a raised bed to keep voles from eating all of your plant's root systems.
And I've never heard of it being used for this, but I would imagine that a layer of river rock would work well to keep the voles out as well.