This picture is looking from the left side of the engine, looking over the battery at the oil dipstick. The yellow handle threads into the engine - you should be careful not to cross thread this when replacing the dipstick. It is a little difficult to navigate the dipstick in and out. Unscrew it, and lift it up. Lower it so that the lower end of the dipstick goes between the engine and the battery. Then tilt it and pull it the rest of the way out. Reverse to put it back in.
Here is the oil dipstick placed on a paper towel. There are markings at 4 and 6 quarts:
Check that the oil level is between 4 and 6 quarts. Keep in mind that the engine is air cooled and that the oil provides a significant amount of the cooling, so you probably want to be a bit generous with the oil on a hot summer day. On the other hand, I've always been told that if you add excessive oil, it gets blown back out very quickly (and this seems to be the case in my experience). Therefore, when temperatures permit, you probably don't want to add oil until it gets all the way down to 4 quarts.
This is the fuel line coming out of the bottom of the main fuel tank, going down to the gascolator.
It's difficult to see much of the gascolator in this photograph, but you can see the blue fuel fittings attached to it. The left side is where the fuel line comes down from the main fuel tank and attaches to the gascolator. The right hand blue fitting attaches to an orange fuel line (just barely visible here) which runs over to the carburetor. Make sure none of these fittings or lines are leaking fuel.
The yellow tube is the gascolator drain. Place the yellow tube in your sampling device and push up to drain some fuel. The gascolator is not a filter. Think of it as a cup with the two fuel lines attached at the top. Fuel flows in one line into the cup, and then out the other fuel line. Materials which are heavier than gasoline will sink to the bottom of the cup, rather than flow out the outlet fuel line. This includes water, but also particulate matter. When you drain the gascolator, you remove the heavy materials from the bottom of the cup.
The other test to perform is to roll the throttle hard against the override spring. Raise the collective al the way. The throttle should just barely move just as the collective hits the top of it's travel. If the throttle linkage either doesn't move at all, or moves more than a tiny fraction of tan inch, the rigging of the throttle may be off.
Make sure the battery and the relay are both securely mounted to the aircraft. In some R22s, the battery is inside the instrument panel. In others it is under the left hand passenger seat. From the Alpha and Beta model on, this location in the engine compartment is typical.
Check the alternator belt for cuts or cracks. Check the tension as demonstrated in the picture: grab the plate and try to rotate the alternator. If you can do it, the belt is too loose.
Just like other places on the aircraft, we want to check where the frames are welded to make sure there are no cracks.
The sheet metal wraps around the engine to conduct cooling air to where it is needed. Check that no cracks are present, and if they are that they have been properly stop drilled by a mechanic. We are not concerned by the valve cover (with "Lycoming" on it) but by the sheet metal around the valve covers.
Here is another picture of some of the sheet metal you should be inspecting for cracks. Look around, there is more of it than you might think at first.
Same as on the left side: grab hold and give it a wiggle to make sure it is properly attached.
Take a step back and look at the overall situation and make sure that the entire engine looks normal.