Travel to a guildie. If that is something you do not want to do then do the main quest line and get some great skill points on the way. Most are quick quests.
Main quest give skill points, big chunks of xp and teaches you, cold harbour was only good for grinding back before one tamriel, now the zone levels as you do so it’s no better than anywhere else
You can join a large guild, then watch who is online in the guild roster. You can port to any online guildie at any time without asking, which can be a fast way to open wayshrines. Just sort by who’s online (green), select a player where you want to go, then travel to.
Coldharbour isn’t my favorite place to grind, honestly. Even fighters guild. It used to be a good place, but now that everything is the same level it’s not really.
On a new character you need skill points as well as leveling, so first do all the public dungeons, which have. A shard and a full point. Do their quests too, which are all self contained in the dungeon and have big xp rewards. For repeat grinding and leveling fighters specifically, a lot of these are undead and Daedra dungeons, and are packed far denser than Coldharbour. Specifically the dungeons in shadowfen, eastmarch, reapers, rivenspire.
But better than anything for fighters, close the dolmens. You get all those Daedra kills, plus the first time you close each dolmen you get a massive boost to fighters. (You’ll see it when the xp comes in, in the lower right corner). On my last couple of new characters I started them by running all public dungeons and dolmens in the vanilla game, probably in training armor during an xp event. They wind up around fighters guild 7-9 when they hit vet, no particularly long grinds other than maybe doing the alikr dolmen Zerg a bit to finish off some psijic.