There's far too many for me to remember so here are a few;
1. Go to Edinburgh see the city and its history and marvel in the crazy accents of the crazy but fun locals.
2. Visit Windsor Castle, you can go around it and it's one of the most beautiful and well kept castles around. The queen goes there a lot, the local area and pubs/restaurants are also great. Once you've seen the Castle you can take a walk along the picturesque canal.
3. Go to the Lake District and see the most amazing scenery and if you're of that sort of ilk camp and enjoy a fire in one of the most beautiful places. Be aware though it's one of the most rainy places in England so if rain and camping isn't you're thing it's worth staying at a B&B or hotel.
4. As others have said, go to a football match. I watched my beloved Stoke lose 2-1 to Chelsea today but it was a good event overall. If you do go to a football match please remember the English love to fight and football grounds are one of the acceptable places to scrap in England. So choose where you sit and who you go to see. There are plenty of scumbags who love to scrap here. But go there with your eyes open and have a great time. I'd suggest not wearing any football shirt if you don't know the area/ground.
You could also come and see the loudest fans in the Premiership http://www.youtube.com/watch?v=T7SqgtuJdbk&feature=related
5. Take in the British love for music. We always have huge amounts festivals and events going on. Liverpool is a good place to visit if you like the likes of the Beatles. I always enjoy stopping by the Beatles Hotel in Liverpool and there’s a museum too.
6. Go to an Old English pub and just watch life go by and have a pint of real ale!
7. Visit Cork in Ireland and take in the quaint Irish ways. I've always found the Irish to be hospitable people and Cork has a history of Glassmaking. If that’s something that interests you, you can take tours and watch them do it. I'm normally not into that stuff but I found it interesting. I find Dublin is not only expensive but very touristy now which takes the charm away for me.
8. Visit what was once the Pottery centre of the world (Stoke-On-Trent) and see the history. You can also buy pottery very cheap from the factory shops.
9. Go to Stone Henge, spend half a day there and then make the short trip to Salisbury Cathedral. The Cathedral had moved my fiancée (Miss K) at how amazing it is. Its architecture is amazing and it also has things like one of the four oldest Magna Carta's in the world. It has Europe’s oldest working clock and was built in the 13th Century. It's truly an amazing place and when you are stood by a knight’s shrine and the incredibly powerful organ starts to play you'll see why people go there from all over Europe.http://www.salisburycathedral.org.uk/
10. Join a Heritage trust like English Heritage Trust for a small amount and then go and explore English history. Loads of places to see and things to do.http://www.english-heritage.org.uk/
There's loads more stuff and I could go on all day!