I'll try not to let this run too long but there's a lot more ground to cover here than you might expect.
Flatten pins in dovetail joints
Plane boards that are about 3-4 feet long maximum, more often than not only about 2 feet
Square boards with a shooting board
Honestly all those things can be accomplished with any bench plane within reason, so from a 3 to a 6 in Stanley's numbering system (and including both a 4 1/2 and 5 1/2).
But in reality you wouldn't normally choose to do all of that with just one plane. Planes no. 3 through to 4 1/2 are intended to be smoothing planes, and they are geared towards doing just that, smoothing. They're not intended for flattening tasks, although they can be pressed into service for that it's not without a penalty in time and effort.
I am probably going to order one Lie-Nielson to get started with.
While you're unlikely to be unhappy with an L-N purchase I'm basically going to recommend straight out that you reconsider and go with something else. For someone in your position I think a cheaper brand or vintage from the secondhand market are a much better proposition.
A premium for what?
My general point is about paying a premium for a plane (and L-N's are among the most expensive of modern planes produced in large numbers, particularly so outside of North America). One of the things that you pay a premium for is higher standards of machining on all parts, with tighter tolerances, which supposedly gives you a plane that works better, adjusts more easily, keeps setting better and lasts longer. But in reality most of that is smoke and mirrors, because planes produced not to be "heirloom tools" but merely as working men's tools — vintage planes from any number of makers, including Millers Falls, Record and Stanley of course — have a proven track record in both performance and longevity, with many planes well in excess of 75 years old still in daily use in a professional setting.
This shows quite clearly that we don't actually need those far higher tolerances for real-world performance.
And of course it calls into question just what the premium pricetag is actually for. Since it's not about performance or about durability feel free to speculate what it is you're actually paying for.
Lower-cost planes can require more work (called fettling, see more on that in the link at bottom) to 'commission' or get ready for use, and the lower price point reflects this. Notice I said can require more work. Not all cheaper planes require anything like the effort to commission that you read about too frequently online (two hours+ needed to lap the sole flat and so on). I've had three planes new planes pass through my hands that each cost less than $30 and an experienced plane user could have gotten any one of them ready to do fine work in under half an hour.
Lastly, one of the other things that buying cheaper allows is buying more tools for the same money (less money in fact), which leads on to my recommendations for you.
What I'd get instead
The important take-home message here is you can get equal performance from planes that cost far less. On top of that for the new user you're less likely to want to baby a tool that didn't cost so much, and if you won't be hesitant to use it hard when needed it can be a great boon.
So in reality what you'd really be better off with here is a selection of planes from a cheaper brand, or, the same selection in vintage tools.
2-3 planes instead of one I would want to have at least a no. 3, 4 or 4 1/2 (pick based on your hand size and strength) and a 5, the latter set up traditionally with a curved cutting edge on the iron for faster stock removal. And given the choice I'd also like a low-angle block plane too, although this isn't at all a must-have and you could quite happily live without one*.
One plane only If you go with just a single plane I would get a 4 or 4 1/2 and a spare iron which you can sharpen differently. This in effect turns the one plane into two different planes depending on which iron is fitted — cambered iron for flattening and faster stock removal, iron sharpened straight (with corners rounded off) or with a very slight radius for smoothing tasks, jointing edges and end-grain work.
A little more on some of these points in these previous Answers:
What makes a high quality bench plane?
Fettling a hand plane
Different ways to set up a Number 4 bench plane
*These days it's often said or implied that you need a low-angle plane for working on end grain, but that's not at all the case. Actually low-angle planes are a relatively modern thing and just as in the past any standard bench plane is perfectly capable of planing end grain if sharpened to a sufficiently high standard. This is why the most common plane used for tackling mitres in a mitre shooting board might be a no. 5 (some users even prefer a 6 or 7 for this, for their extra heft).