You'll be in a nice part of the country, not too far from us
. Bognor Regis is a traditional seaside town but has gone downhill since it's glory days but just outside of it, where you will be, is some glorious countryside. Nearby Chichester is an upmarket medieval town with plenty of old historic buildings and monuments to see. There is also the Fishbourne Roman Palace nearby which is excellent if you are interested in Roman history,
https://sussexpast.co.uk/properties-to-discover/fishbourne-roman-palace
Bognor Regis to Stonehenge will only be a 1.5 hour drive on mainly good, fast roads if you do want to see it, Highclere is 1hr 20 mins. If you leave after the rush hour then the journey should be pretty uneventful however the A34 at rush hour can be a nightmare likewise the M27. The train from Bognor to London Victoria is less than 2 hours.
Going the other direction Brighton is just over an hours drive. Plenty to see there and the place has a nice atmosphere. A bit further is Hastings where you can visit Battle Abbey and the site of the Battle of Hastings, you can drive there but I would prefer to take the train from Bognor.
Also within close reach is Winchester, the ancient capital of England where, amongst other great sights, is King Arthur's Round Table
http://www3.hants.gov.uk/greathall.
The great city of Portsmouth
is a 50 minute drive from Bognor where Charles Dickens was born (you can visit his birthplace) and the Historic Dockyard houses some magnificent ships such as HMS Warrior, HMS Victory and the Mary Rose. The new Mary Rose museum is a fantastic exhibition and well worth visiting
http://www.historicdockyard.co.uk/. If you have a look on Groupon there are often discounted tickets for the dockyard.
I also love Portchester Castle, on the outskirts of Portsmouth overlooking the harbour, a small but well preserved Norman (mostly) castle.
http://www.english-heritage.org.uk/visit/places/portchester-castle/
From Portsmouth (or Southampton) you can take the ferry or hovercraft to the Isle Of Wight. Osborne House (Queen Victoria's holiday home) is a great attraction and is more interesting than Highclere in my opinion (unless your sole reason for visiting is Downton Abbey)
http://www.english-heritage.org.uk/visit/places/osborne/
Depending on your interests and what you intend visiting it may be worth investing in an English Heritage pass for overseas visitors, a 9 day 2 adult pass costs £52 and will grant entry to all English Heritage sights for no additional payment
http://www.english-heritage.org.uk/visit/overseas-visitors/. Stonehenge, Osborne House and Portchester Castle are all English Heritage sights so visiting those three alone will more than recoup the cost of the pass. Here's a list of the English Heritage sites closest to Bognor Regis
http://www.english-heritage.org.uk/visit/places/#?category=Property&theme=&page=1&place=bognor regis
I also like Arundel Castle which is a 20 minute drive from Bognor
http://www.arundelcastle.org/
If you like walking then the South Downs has many walking trails and many cosy pubs to finish up in
https://www.southdowns.gov.uk/enjoy/explore/walking/
A 2 hour drive west will take you to the Purbeck coast in Dorset (en route you will travel through the picturesque New Forest national park) which is known as the Jurassic Coast due to the abundance of fossils there. There are some fantastic walks there along with quaint villages and traditional pubs.
There is so much to see and do in the area it's impossible to list them all, the ones I've highlighted are a few of the first that came to mind. Of course a lot depends on your interests but I have no doubt that you'll easily fill your time. Hiring a car is definitely recommended, whilst most sites can be visited by public transport it can be quite time consuming and the costs soon mount up. A small car should be relatively inexpensive (manual) but fuel is not cheap however a small fuel efficient car will keep the costs down.
As for the weather, late October is variable. We could have storms coming in from the Atlantic, it could be sunny and cool or we could have a late Indian summer. You're from Ohio.....it won't be cold. The average high is 15°c and low is 10°c, rain is a distinct possibility, the drizzly type rather than the pelting type. Snow is highly unlikely. I'm still in shorts in October!