I live here (was in NYC for 5 years, now just outside of it for the last 2 years)
Shopping - tbh, 5th avenue etc is worth a look, but there's nothing out of the ordinary that you now can't find in the UK, unless you want specific US designers, makeup etc that might be hard to find at home. I find it expensive, especially for decent shoes - a lot of it will depend on what the exchange rate is when you come
Eating out - we eat out a lot here, usual price in a nice (not posh, just nice non chain) restaurant for 2 people with 2 courses and a bottle of wine is anywhere from $100 upwards and then tax and tip on top. There's obviously lots of places that are cheaper than this and you can get good deals - open table is a great resource for bookings and suggestions. There's also a ton of decent places for lunch places - build your own salad places and stuff are v popular, and its really easy to find gluten / dairy free or for vegans catered pretty much everywhere
Stuff to do - I think its worth looking at those bundled tourist passes they do for museums etc, as paying as you go can be very expensive (its about $20 each for MOMA etc) ellis island is worth a visit, seconding staten island ferry for a cheap way to see statue of liberty (just don't get off and actually go onto staten island, its an odd place. Take the first ferry back!)
Walking across brooklyn bridge is a good one (just go early as it gets rammed) and then DUMBO on the brooklyn side is great for a wander. Brooklyn Heights has a great park, and is also one of the oldest neighborhoods and worth a visit.
I love MOMA for its collection, the guggenheim is a great building but I never really rate the art there. Central Park is ok - its bloody massive tho and really easy to get lost and really its just a park (so shoot me...) I'd also recommend doing stuff like the tenement or transport museum if that's your thing.
Downtown where the world trade center and financial district is very very popular (and busy) but there's not really much to see, which sounds odd I know. You can't get near the stock exchange etc for security reasons, the WTC is a memorial but I find it quite disconcerting the amount of posing selfies that go on, given what happened there (personal opinion tho) Going up the top is a nice visit, but, like the top of the rock etc, it can be difficult to see what you're looking at.
Definitely narrow down what you're interested in and google discounts before you come, as there's always stuff on the cheap.
Get comfortable with the concept of tipping before you come, it can be eye watering, but the service is usually better as a result. $1-2 for each round of 'normal' drinks in a bar, slightly more if you get cocktails, 20% for meals out (i.e. where you get served at your table) and do this before they add the tax, 10% for taxis. Lots of coffee bars where you order up front etc have started being cheeky fuckers and putting a tip jar by the tills, but I don't tip unless its my regular place