Hi there. It's great that you are getting into the comics. I would highly recommend the 'X-Men: Dark Phoenix Saga' trade paperback which I feel is the best X-Men story. You can find it online at Amazon, Book Depository etc. The "Epic Collections" are a cheap way to get a stack of comics. The book "X-Men Epic Collection: Second Genesis" comes out in April 2017 and begins Chris Claremont's iconic run.
Don't worry about the comics being old. They are classics.
This version with Scott holding Jean is the one to get --- it includes 138 and some extra stories, and it's cheaper right now on Amazon too.
If you have some extra cash I highly recommend Proteus, which is the prologue to Dark Phoenix and a really classic story too. It will help you understand what's going on in Dark Phoenix -- it happens right before.
If you have $50 to blow the most complete collection will be * Marvel Masterworks: The Uncanny X-Men Volume 4 & 5.*
Claremont and Byrne? According to the Amazon listing it collects X-Men 129-138, Classic X-Men 43, Bizarre Adventures (1st Story), and Phoenix The Untold Story.
https://www.amazon.co.uk/dp/0785164219/ref=cm_sw_r_cp_api_i_yvVcDb01FWC5J
I would suggest all new x-men and the current run on uncanny, as it's reasonably simple, but it's quite cyclops-centric (even though I really enjoy it).
Maybe try Astonishing X-men by Joss Whedon. Thats pretty simple, and though he Cyclops does feature heavily the team dynamic is balanced, so you don't get too much of one character.
If you want really easy to follow classic stuff try early era Chris Claremont stuff.
So looked over the adamantium collection, and it’s not bad. It’s kind of a greatest hits but it jumps around 40 years worth of stories and a lot of them have lead up/follow up issues that are missing. So for that price too, you could probably get him a bunch of complete storylines. Not sure what your price limit is but seeing as the adamantium collection is between $200-400, you might try the following
Has the Brood Saga, The Frank Miller Wolverine, God Loves Man Kills, and others.
You could get both of those for a little over $100 and buy another collection for the price of the AC. However, if he likes rare and hard to fine stuff, it’s kind of a cool thing to have on the shelf. Good luck!
The thing about comics is that there are big stories and small stories, and stories within stories. A story can be complete in itself, but there might be optional extra supplemental material for that story.
For instance, when it comes to the Phoenix Saga, you could get this collection: The Dark Phoenix Saga, which is a complete story in itself, but the story in that collection starts with Jean already having become the Phoenix. If you want to go all the way back to when Jean became the Phoenix all the way through to the end of the Dark Phoenix Saga, there's this: X-Men: Dark Phoenix Saga Omnibus, which includes a lot more material and has basically everything you could ever want to read about that story arc.
And just so you know, Marvel also has a digital comics subscription service called Marvel Unlimited, where you can read that whole story for a good amount cheaper than buying a physical book.