A friend asked me about making kid's pants. We found a pretty good tutorial that helped a lot. First, I make a pattern from a pair of pant that fits well, adding about 1/2" seam allowance, and tracing the pants onto the inside of a paper grocery sack. Add about 1 1/2" to the waist to allow for folding, elastic and stitching. Same for the hem.
Both panels are roughly the same, so I use the one piece, putting the straight edge on the fold of the fabric. It's important to fold the selvedge edges (usually with words on them) together, and use the fold in that direction, so your pattern is running in the same direction as the grain of the fabric. (This is especially important if you are using corduroy or velvet - all patterns should be cut the same way.)
Next, to sew the pants together, you can start with the legs, sewing right sides of the fabric together.
Or you can start with the inseam, sewing the two pieces together. It's a good idea to stitch this again, a little closer to the edge, just to reinforce it.
For the elastic casing, fold over the top about 1/4", and then again about 1/2 or so, press and stitch. Or serge the end, fold and stitch. Leave open about an inch so you can fit the elastic in.
Same thing for the hems.
One big question I keep seeing is about how much elastic you should use in your pants. A rough guide is about 3/4 the measurement of the waistband, maybe a little less if the elastic is quite stretchy. This pair used 1/4" elastic and tends to come down even over a cloth diaper, so I'd use 1/2" or 1" elastic next time.
Put a large safety pin through the elastic (as large as your casing will allow). Thread through casing, remove pin, overlap the ends about an inch and stitch elastic. Stitch casing closed.