The runway numbering scheme is to take the nearest magnetic heading of the runway, round to the nearest 10 degree increment, and drop the final 0. This means that a runway which has a heading of 224 degrees will usually be called runway 22. If there are two runways with the same orientation, then they will usually be called L & R as in 22L and 22R. There are even some airports where there are three runways on the same orientation, and then they have a L, R, and C designation. The opposite end of the runway, has the reciprocal number. In the example above, 22L becomes 4R at the other end.
The plot thickens when there are 4 runways with the same orientation such as DFW which has 5 runways that are all oriented to 175.4/355.4 degrees. In this case, they are split into two different sets of numbers. At DFW, we have 17L, 17C, 17R, 35L, 35C, 35R on one side of the airport, and 18L, 18R, 36L and 36R on the other.
As for taxiways, they are lettered. At most airports, the taxiways have single designations such a A (alpha), B (bravo), etc. As some airports, the taxiways will have two letters or a letter and a number such as SC (sierra charlie) at IAH, or L4 (lima 4) at DFW. In the case of IAH, the S means that the taxiway is on the South side of the airport; on the North side they begin with an N. Interestingly enough, in the Northwest corner, there is a little confusion because the N taxiways meet the W taxiways. In DFW, L4 means that it is a taxiway that comes off of Taxiway L, and is between L3 and L5.
In the old days, ORD had real interesting names for taxiways such as "scenic," "stub," "new scenic," etc. I suspect that some near calamity caused by the non-standard markings, an FAA directive, or safety audit, etc., brought about the change to standard markings.
To view some airports, just type the airport identifier followed by "airport diagram" into Google.