Avoid Worthing: It's a car park at the best of times
& if heading West Worthing to Arundel, check the jam at the West end of the dual carriageway: it's best avoided
As a rule of thumb, don't take the coast road unless you need to. Take a road one or better two in from the coast, and you can find old A & B roads that pass through quiet leafy scenery.
Yes, Will got there before me. As well as Arundel you want to avoid the A27 Chichester bypass completely, especially on a sunny weekend (I stupidly tried to go from Chichester to Havant a couple of weeks ago and the 14-mile journey took me over 90 minutes). Goodwood is nice to visit if you've never been but be aware that according to the
Motor Circuit Diary there is a running race on that day so don't expect any fast action. I doubt the house and estate will be accessible because the build-up for the Festival of Speed will be going on. If you want to go from Goodwood to Tangmere then cut through Boxgrove.