Clay bar, yes absolutely. Think of all the years of junk you have accumulated on the surface. Washing it won't remove the grit and contamiments that you pick up in driving. Clay bar will strip it off and leave a smooth finish that you can then start improving by removing swirls, polishing then putting on your wax. And it really doesn't take that long. I do it twice a year, don't even think about the effort, which is well worth it. And my original paint is 16 years old but most people think I've had it repainted. I'm not a waxer, but I know what great results I've seen.
For the engine, try Simple Green. If really dirty, use the blue colored aviation stuff, really cuts grease and grime. If not real bad, the regular green colored Simple Green, and a lot of scrubbing will work fine. Use some 00 or 000 grade steel wool on your shiny stuff under the hood, then a good chrome/SS cleaner like Mother's, etc., you'll be happy with the results.