Exploring SE England from Norwich?

Norwich Cathedral is wonderful and the city itself fascinating and ancient, with more than enough to occupy you for several days. It's a compact, attractive and extremely walkable place and...of course..it is perfectly safe to explore on foot. 
Make sure you visit the truly superb Medieval cathedral:
https://www.cathedral.org.uk/
and the Norman castle & museum:
https://www.museums.norfolk.gov.uk/norwich-castle
There are other museums and galleries too, of course. The official tourist info site has details:
https://www.visitnorwich.co.uk/explore-and-experience/city-of-culture/galleries-and-museums/
Explore the ancient streets (Norwich retains much of its Medieval layout and has many historical buildings), walk along the river Wensum and visit some or all of the 33 ancient churches within the city's historical centre. There are also several theatres and, of course, cinemas...and the official tourist info site will tell you what special events are happening during your visit:
https://www.visitnorwich.co.uk/whats-happening/
The city has excellent shopping, with a good market, plenty of independent shops and its own long-standing department store (Jarrolds) as well as the usual UK chainstores. It's also excellent for eating-out. As in all UK cities and towns you'll find a vast range of cuisines from across the world , again with plenty of independents as well as larger chains. You'll find places to eat throughout the city. Tombland (the area just outside the cathedral precincts...nothing to do with graves.... its name is Saxon for 'open space'!) is a particularly good area to explore unless you're on a very tight budget indeed.
You'll find food to suit all budgets but if you're looking for good beer and decent food at very reasonable prices try the Wetherspoons pubs in the city centre: the Glass House on Wensum Street, the Bell Hotel on Orford Hill (near the castle) and the Queen of Iceni in the Riverside Retail Park near Norwich train station.Pubs in general tend to be good places to find decent food at reasonable prices, though quite a few Norwich pubs are 'bistro' pubs and thus tend to be pricier.
In common with all UK cities and towns, parking in the city centre is neither easy nor particularly cheap so if you're just going for the day it's best to take bus or train if possible.
Day trips by car:
Seaside Cromer is famous for its crab (have a sandwich!) and the coast between Cromer and Hunstanton is wonderful for nature (especially birds) and walking (it's an Area of Outstanding National Beauty).  http://www.norfolkcoastaonb.org.uk/activemap/
If you're looking to experience a 'traditional' (and rather tacky imo) UK seaside town, with rides, candy-floss, b&bs, pubs, bars, clubs, shops and miles of sand try Great Yarmouth.
Charming, historical Bury St. Edmunds is well worth a visit for its shopping, its food and its ancient abbey ruins ...and perhaps a tour of the Greene King Brewery? :-) https://www.visit-burystedmunds.co.uk/
West Stow Anglo-Saxon Village is fascinating, will give you a good  idea of life in England before the Norman invasion of 1066...and, in many ways, for centuries afterwards as well: https://www.weststow.org/
Southwold is a lovely and unspoiled seaside town (again, in an Area of Outstanding Beauty). Try some Adnam's beer (the brewery was founded in Southwold, walk along the beach and...if you're lucky..a ferryman will row you over the river to pretty Walberswick, long popular with artists. :-)  http://www.visitsouthwold.co.uk/
There are far, far too many other potential days out by car (or train) from Norwich to list here...stately homes such as Blickling Hall,  nature reserves, ancient churches,superb countryside and coastal walking etc etc. You certainly won't be short of somewhere to go. The official Norfolk tourist info site has many suggestions:
https://www.visitnorfolk.co.uk/things-to-do/
And that's just Norfolk....train or a longer car journey will get you to many other fascinating and lovely places in adjoining counties such as Cambridge (England)
Enjoy your stay in Norwich....it's a lovely city! :-)