Hi I’ve been to Badminton for the past 3 years. In a nutshell, it’s amazing and a lot of fun. #1 is splurge and get the membership passes for all 4 days. It’s very economical and let’s face it, its nice to be in a covered area should the skies crack open. Its quite affordable as the membership does not include any VIP perks like Kentucky or MD 5*, however, you can upgrade to the Portcullis Club. The way the arena is set up you can see dressage, some XC and show jumping from the membership tent. No tickets needed for grandstand seats for dressage, but yes for SJ - those sell out fast.
Accommodations: Badminton is in a pretty rural area. Chipping Sodbury and Wooten Under Edge are pretty close as is Tetbury. Last few years we’ve been staying a country hotel about 15 minute drive. A kind taxi driver gave us the “best route” to avoid traffic, it’s kind of a close our eyes and drive haha. The best bet for accommodations is to check out the Badminton website- they have a link for accommodations which also shows the distances to/from Badminton - it includes hotels and B&Bs. They also have a glamping option - which has a bar/restaurant nearby - great atmosphere and food too. I think there might be camping hook-ups if that is something you wanted to look into as well.
The first year, we picked up rental at Heathrow, now we taxi to our hotel, and pick up rental in Chipping Sodbury - I think one year they were able to bring car to our hotel, and we return at Heathrow. Depending on flights - you could fly into Bristol and pick up rental there.
Oh - Bath is also an option to stay and drive to Badminton. Bath is well worth a visit anyway.L Last year we went to Bath after Badminton and stayed at a Doubletree Hotel, not a charming old hotel but excellent location, parking next door, great lounge area and restaurant, and could walk to the Roman baths and other popular Bath sites.