If I want to look through the mysql logs for things that would tell me my server can't handle the load of traffic then what could I search for? Are there any other bad things I should look for in my logs?
There are several things you need to look for e.g. look at server load, site load time, enable slow query log to look for slow queries, look at PROCESSLIST to see how many queries are in LOCKED state and why they are in being LOCKED etc.