If you want to go direct next summer, BA go from LGW and VA go from LHR, so there is that consideration if one airport is much better for you than the other. Otherwise, somewhere like
www.skyscanner.net will give you are fair guide as to prices for your intended dates. Prices are highly variable, bordering on insane for some dates right now. There are 'sales' currently on, not really that focussed on the summer, and there are usually Easter sales at least, if not one other sale window before.
As for hotel, it will depend on where you want to be located and what you might want to spend most of your time doing. 3 weeks is going to give you a fairly hefty hotel bill wherever you stay and saving $20 a night will add up to a decent amount. However, the right way to do it is to stay at a much better hotel for $20 per night less than the advertised rate for the inferior hotel. That's where this board comes in handy, plenty of people have their own fav websites to go to ~ the thing to remember is that the hotel rates aren't fixed, they are highly variable and often fluctuate wildly according to anticipated occupancy. In other words, you can look once and see a bad price, but it doesn't mean you couldn't get a good price for the same hotel via a different route that same day, OR that the price won't be much better from the same route the following day. It really is down to how much effort you want to put in to securing a great deal.
1. Do you want to stay near the poker action at the Rio (offstrip) Or downtown? Or on the strip? Will you rent a car? Usually the best way to get help from this thread is to have a search and put up the dates/prices that you have found that look reasonable to you and others will usually beat it (better hotel, same money or same hotel, better money, and sometimes better hotel for better money)
2. Direct websites are usually best if you are unsure about the length of your stay as you will have cancellation flexibility that 3rd party websites don't offer
3. Are you open to doing things like hotwire / airbnb / vrbo / homeaway / priceline
Usually each year, someone pays more to stay in Excalibur at some point than it would have been to have stayed at Bellagio. Presumably blissfully unaware that they have done so, of course, but that is all part of doing Vegas right.