What is a good place to begin building a kit/set up for performing your own service?
- oil changes
What is the best starting equipment one can start on?
I'm going to answer the basic mechanical points, as weighted by my opinion of importance. I'm leaving out the issue of waxing as potentially too broad.
Safety: These are critical. Do not proceed down the list without addressing each (at least).
Basic tools: These are tools that you will appreciate having.
Convenience: These are things that you will likely appreciate.
First off, everything Bob said.
Socket Extensions: Your socket set may come with 1 or 2 extensions but I'd buy a couple more. Having different length extensions are invaluable for getting to hard to reach items, plus you can combine them together for a longer extension.
Socket U-Joint Adapter: I never see these in socket sets and you will regret not having 1 or 2 in combination with the extensions. The U-Joints allow you to use a socket wrench on a fastener that you can't get a straight line on. It allows you to gain access from more than a 45 degree off angle. Depending on where you put the U-Joint in the chain also alters the access angles heavily.
Combination Wrench Set: Sockets are great for speed but sometimes you just can't get onto a bolt with them and need to use a combination wrench. The box end (end with the fully enclosed circle) is useful for applying more torque, the open end being useful for slipping onto the bolt easier.
Magnetic Pickup Tool: You are guaranteed to drop a screw, nut, bolt, etc somewhere that you cannot get your fingers. These are telescoping or snake-necked tools that you can work your way to the item and grab it.
For Oil Change
Filter Wrench: Some filters (like my K&N) do have a standard nut on the end of it so you don't need to use a filter wrench but for the most part, you need one to loosen the oil filter. Sometimes they are a strap wrench that is a rubber strap with a handle, you wrap the strap around the filter and through the handle, then turn. Others are a round piece of metal that is tightened onto the filter.
Drain Pan/Bucket: You'll need something to catch the oil in. There are ones with spouts in them which are nice to keep from spilling when transferring into your storage container. Still, you can pull it off with just a simple bucket.
Storage Container: After you drain the oil, you need a sealing container to transfer the oil to your nearest disposal location.
For Brakes/Clutch/Other Lines and Bleeders
Flare Wrenches/Line Wrenches: If you are going to be wrenching on hydraulic lines you MUST get these wrenches or you will round them off. If you are only doing bleeding, you can get away with the box end of a combination wrench as you can put it on the bleeder valve before attaching the drain tube and just keep it on as you open and close the valve. See this question too How important is it to use a real flare nut wrench when removing the fuel line on an old Ford tractor?
One Man Bleeder: If you are going to be bleeding by yourself, you need a bleeding tool. If you will have a helper, you don't.
Waxing/Detailing: I can offer you a set of videos recorded by a member of my car club when we went to a professional car detailer (he has detailed cars for some of our members before shows in which they've won some very difficult to win awards) for some instruction.
Following on from Bob and ManiacZX's answers:
A good assortment of screwdrivers are essential - including torx or hex bits if your car uses them. Some socket sets include screwdriver bits, which can be very useful. An old long-handled flat driver can often double up as an impromptu pry-bar too...
Like the above, you'll probably already have a claw hammer in your household toolkit, but a small club hammer can often be useful for shock-loosening stuck bolts (especially on older cars where things have started to sieze)
Keep the following in stock:
Most of those you'll only need to keep in topping-up quantities, buy bigger amounts when it comes to time to change them.
These are car-specific, so you'll need to check what you need. In particular, check what you need to remove the oil drain plug - some cars are a simple bolt that you can put a socket on, others use a square one that needs a special tool. If you don't have one yet, buy a workshop manual for your car, and read the section for the job you're about to do through twice before you even open your toolbox.
Buy the best tools you can afford. Most amateur mechanics build up their toolkits over a number of years, gradually adding new things as and when they need them.