What is a good place to begin building a kit/set up for performing your own service?
- oil changes
What are the best tools/equipment to start with?
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).
Safety glasses: Always wear them, especially when you don't think that you need to. I purchased mine from the local super hardware store for a quite low price. They're comfortable and not unfashionable.
Reliable jack and (4) jack stands: You should never lift your car with a bad lifting tool. I have a fairly expensive one that I really like because I liked the feature set and had some tax refund burning a hole in my pocket. A jack that is rated for less than two tons is not safe to use on any car that I would normally see in the US. Likewise, each jack stand should be well made, reliable and also rated for at least two tons. Remember, if you lift your car and get under it, you're under something that wants to kill you.
Wheel chocks: You should never lift your car without immobilizing the wheels. Putting the car in park and using the parking brake are also necessary but are not sufficient. If those wheels move, the car could roll off the jack stand and, again, kill you dead.
Basic tools: These are tools that you will appreciate having.
Socket set: I have a nice socket set with a variety of sizes that I purchased from a local auto supply store. It has a half-inch driver so I feel very comfortable putting a lot of my (modest) muscle into it.
Torque wrench: Anything that you disconnect on the car must be reconnected. They almost always need to be reconnected to a particular torque. You'll want your torque wrench to fit your socket set. Otherwise, frustration will ensue.
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.
Detailing Tech Session Videos
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.
One more piece of advice, plan a space to collect your tools. It could be as simple as a small tool box or it could be the beginnings of a workbench and storage wall in a garage. Either way, having your tools organized will make the work more pleasant and efficient and will help you keep track of your growing investment. This question, How To Get Organized Without a Garage Workspace?, has some suggestions.