IAWoodsman and Ray Mears videos are the best place to start. Ray Mears even has a video on this exact issue, what to look for in a well rounded axe collection. Also WoodTrekker is a fantastic blog written by an experienced bushcrafter who has a thing for axes. Lots of good beginner info on design, construction, use and maintenance. Gransfors has a downloadable book on the history, design and maintenance of axes called The Axe Book, also a required read. I'll start with a few suggestions, some 'classics'.
American Felling Axe 31-35 inches, 3-4.5 pound bit-Classic simple Jersey pattern axe for felling trees. Curved hickory haft for shock absorption, ergonomics and maximum leverage. Lots of classic companies out there, Kelly, Plumb, Mann/Norlund, Gransfors even makes a nice one
Hudson Bay Axe-hatchet size-28 inches, lighter bit-classic northern design featuring a wide bit that trims back significantly to the eye, reducing weight and maximizing effectiveness for packing backcountry. Norlund, Snow/Nealley and Council tool seem to be the best/most classic manufacturers, though these days Snow and Nealley less so.
Double Bit Cruiser axe-28 inches, 2.5-3 pound bit-classic horse packing and canoeing axe, one thin bit for deep biting and one thicker convexed bit for knots, roots, frozen wood, etc. Kelly makes a really nice US built contemporary one, though there are many classic companies. Norlund cruisers are highly sought after.
Swedish Carpenter's axe- 20 inches, 2lb head-specifically designed for carving wood, with a nice thin flat bit. Great for bushcrafting and making furniture, bowls, dugout canoes, all sorts of stuff. Husqvarna, Wetterlings and GB make them.
Tasmanian style Competition Work axe-short haft and heavy head designed specifically for racing, though they tend to be super efficient felling axes. Couple of obscure/specialized companies make them, though you can find an Arvika or Helko factory new for around a hundred dollars.
Scandinavian Forest axe-20-26 inch, 1.5-2.25 lb head-thin profile blade made with Swedish axe steel-incredible allaround packing-sized axes for felling, splitting, carving etc- the one-in-all, do-it-all axe as produced by a number of Scandinavian companies. Husqvarna, Hultafors, Sandvik, GB, Wetterlings, SuperBanko, Arvika (I know I'm overlapping factories here....)
Tomahawk-light do-it-all survival design, good for everything from fighting to throwing to wood processing to bushcrafting. Lighter contour knife-blade style bit that tends to be fast in the hand and very well balanced. 18-24 inch straight haft is common. HB Forge makes a great one, though the Cold Steel Frontier or Pipe Hawk wouldn't be a bad user to start with.
Here are some possibilities I'd consider in a well-grouped axe collection. Still working on some of them myself, though I've ordered many this week

Current collection consists of:
Jersey-style Swiss surplus hatchet
Norlund Hudson Bay 14 inch hatchet
Sandvik 14 inch scandinavian forest axe style hatchet
Husqvarna 20 inch scandinavian forest axe
Plumb 20 inch carpenter's axe
Wetterlings 26 inch scandinavian forest axe
Wetterlings 31 inch scandinavian forest axe
HB Forge Shawnee hawk
Cold Steel trail, norse and frontier hawks
Cold steel trench hawk....
probably missing some. I have many on order too, will double my collection in the next two weeks. If you stay active in the community, axe lovers are really into their hobby and like to share it with others. I've picked up on and made the most of some amazing deals I found out about on here. I paid 20 dollars for my 20 inch SFA, which would typically retail about 80 bucks, for example. There are other sources that pay to be diligent about. I got my 31 inch Wetterlings axe for 10 dollars at a garage sale, and sunk some money into a new haft for it, but basically they retail for 90+ dollars. Knowing quality brands is important.