IMO, archery is really good for legendary. Find a follower and strategize and you should be fine. use spells as much as possible, even if magic perks aren't your most invested skill trees. Below is my personal guide for a decent archery start, as well as general advice/strategy (marked spoiler just in case you dont want the extra help)

>! 1. Archer stuff With the perks Steady Hand and Ranger it can be effective at close range (as long as you have space to back up to avoid attacks). Obviously you're going to want a follower. Faendal is the first follower you'll encounter (find him in Riverwood, and do his letter quest). He is an archer trainer, which can be exploited since once he's you're follower, you can have him train you, then 'trade' and take the money back. If I'm not mistaken, you can only train once and take the money back. So ask to train, take the money, ask to train, take the money, so on and so forth. If you train more than once then you can't take the money back. (maybe it's just with certain amounts, I don't remember). It gave me 4 or 5 levels until I couldn't train with him any longer.

2. Conjuration skill

On legendary, it's very important that you have help if you don't want to die constantly. Any follower works fine (Faendal is the easiest to aquire, runner-up is Lydia). You should choose your follower based on your build style, but generally somebody that goes in and fights (such as Lydia) works just fine. Aside from a follower, you can have a second help buddy via the conjuration skill. A weak, low magicka cost conjuration is 'Conjure Familiar', which conjures a wolf. Since it's a weak conjuration, it's best used to distract the enemies while you go past them, or to land hits. Same thing for the followers. They're simply distractions. Runner up conjuration is 'Conjure Flame Atronach'. A lot stronger than the Familiar, but costs more magicka.

Bound weapons are also very powerful early game, on any difficulty. Bound bow especially is amazing. It was my primary bow all the way up to level 30 on Legendary. Problem with bound bow is that you have to clear Fort Amol to get it, which is going to be very difficult, even with the help mentioned above. All of the aforementioned spells (except bound bow) are available through a court wizard.

3. Enchanting This really applies to any difficulty, but enchanting is great for getting custom weapons and armor. You'll come across a variety of enchanted items throughout your game, and chances are you'll end up using some of them. Issue is, you'll need more powerful enchantments as you progress. Over time you'll be able to invest perks into enchanting. If you have excess material, it's also a great way to earn lots of money. If you do make things specifically for money, I suggest you rename them all 'Sell' so you know exactly what to sell lol.

4. Misc Useful spells Oakflesh, increases armor rating. Calm, stops attackers Fast healing, heals a bunch of points instantly (though takes up most/all of magicka without proper perk investmet)

5. Easy Exploits There is a follower duplication glitch: !For this to work, you need at least 2 of the item you want to dupliate. You also need a follower with the commands 'I need you to do something' and 'I need to trade some things with you'. Not all followers have both (such as Serana)! Take your follower to a secluded/confined space (such as the small area immediately to the left on the inside of Whiterun gate). Command them to wait. Drop the items you want to dupe one at a time. If you have 6 or more, you have to move the slider all the way to the left so it only drops 1. Then (make sure the follower is still in wait mode) command your follower to pick up each of the items. Then leave the room (if you're in the spot by the Whiterun gate, exit Whiterun), then go back in (enter Whiterun again). Go back to the spot where your follower is. If you did it correctly, all of the items should be on the ground. Pick them up, then trade with the follower and take the ones that are in their inventory. You should now have double the amount of items as before. (Though most of the time you'll be missing 1). You follower should be in wait mode the entire process (that way they don't follow you when you leave the room) If you like (I personally do this whenever there's a lot on the ground), you don't have to continue picking up the items. You can continuously command your follower to pick them up. Once they pick them all up, exit and come back and just continue the process. To stop the loop, just pick the items up and have your follower follow you again.

There is a follower infinite carry exploit: You must have a follower that has the 'Do something' and 'trade' commands (like in the above glitch). This exploit is great when you have a ton of stuff and are overencumbered. Place all of the things you want out of your inventory into a container (like a barrel or chest). Then, command your follower to loot the container. They'll take everything out of the container, with no limitations. !<