I think this is a great question.
The two basic options are make it yourself or have someone else make it. EDIT: check the bottom for a third option...
Make it yourself
To make it yourself you need to choose a material and manufacturing process for the part, then acquire the material and skills needed. (Usually people select a process they know how to do already.)
For a hobby robot, often the exact material is not too important. For instance, for the part you show, you could make it from wood, plastic, or metal and it would work ok. Cardboard or corrugated plastic (for making posters) would work too and you could cut it with a hobby/xacto knife.
The way to go is to find a friend that likes to make things and then get their help to learn some making skills. You can also try learning from a website or book. (another website)
Have someone else make it
There are some companies that focus on the hobby market and you may find their services the easiest to use. Pololu and eMachineShop are two of many. I have never used them so you should read reviews online to determine if they will work for you.
There are lots of companies that specialize in fabricating parts using one process or another. You would have to learn a little bit about each process to choose whether it will work for any given part. If you are only making a few parts it is usually too expensive for a hobby project. eMachineShop is an easy way to get a general idea of what the cost of different processes because you can create your shape and then ask for quotes for different processes to make that shape. Laser cutting vs machining for the circular plate for instance.
If you can afford to use eMachineShop, it's definately worth checking out your local shops. The two factors that determine the cost are the quantity of parts and how "hungry" the shop is, meaning, how much business do they have going on right now. It's harder work to contact a lot of local shops, but if you do, you will probably be able to find one that will be cheaper than eMachineShop.
For simple shapes there are companies that have a catalog of basic designs for which you can change some dimensions or details when you order. Misumi is a good example. For basic shapes like shafts, brackets, bearing blocks and spacers, there is a good chance that you can get a semi-custom part cheaper than if you had a shop make it.
Option 3 - Adapt something that is close
There is no rule that you have to start from raw, unprocessed, material when making something. You can use something you find in a junk yard or at a store that is close to what you want and make small changes to it. Also, very few robots are made entirely from scratch. They all use components like bearings, motors, couplings, gears, and other things. There are many sources for these components. In the US, McMaster Carr is a good place to start, RS is an international company worth checking.