you also have f3/gt at donington on the 22/23rd, which is a decent opportunity to visit the place. it's good for viewing and obviously has a bit of history. not sure i'd call it unmissable, but the british gt grid is looking like it's going to be another packed one with a decent amount of variation if you're not bored of sportscars after le mans! there's something 6-7th july at brands that includes the porsche club gb, which is a club level series. not a big event but if you'd like to see brands then that'll be a good option.
i'd recommend the touring cars at oulton as mentioned above by toledo47 for the following weekend - it's a lovely scenic circuit, and the viewing isn't bad there as well. iirc racing doesn't start there until midday on the sunday, which is something to consider if you were hoping to see a lot of track action.
as for museums, perhaps someone else can help but the
national motor museum at beaulieu may be worth a visit, and if you're into general history, the imperial war museums in london and at duxford (for the planes and concorde) are absolutely must-sees.