Leveling combat is done by killing things, bestiary, slayers, dungeons, and minions.

Early on, you'll find running around spiders den, doing low level slayers and completely bestiary milestones one of the more effective ways of leveling combat - particularly on a budget. But once you start reaching level requirements such as level 12 for the End Island you'll have access to much better gear like the Ender set, Dragon Set(s) and start using the best utility weapon in the game - Aspect of the End. Pretty much, once you reach The End try and pick up one of these Dragon Armor sets as it's going to keep you going for quite some time. An older money making and combat XP method is running around the dragon nest with Wise/Young Dragon Armour right clicking Zealots with a Aspect of the Dragon - it's super effective and not a bad way to make some coin.

The reason why bestiary is so good is every second level is a combat XP reward. After Combat 18 every second level (even numbers) is 1,000,000 XP and to level Bestiary, you just hit milestones by killing a certain amount of a mob or mob type.
wiki.hypixel.net/Bestiary

Slayers, access by Maddox in the Hub, is another great way to gain XP. You call up Maddox on his batphone or visit him and select what tier of boss you want to do. They're currently split up into 5 types, Zombie (Revenant Horror), Spider ( Tarantula Broodfather), Wolf (Sven Packmaster), Enderman (Voidgloom Seraph) and Blaze (Inferno Demonlord) each with 4-5 tiers in each which all drastically increase in difficulty. Start off with the first 3 as there's some pretty important items you unlock such as the Maddox Batphone, Mana Orb and auto-slayer which helps you chain complete a load of slayers without having to contact Maddox every time.
wiki.hypixel.net/Slayer

Dungeons which is accessed through the Dungeon Hub, has 7 different floors and two difficulties. Each floor consists of secrets, mini bosses, puzzles and a boss room at the end. Upon completing the dungeon you'll be rewarded with a good amount of XP and a choice of chests which are unlocked by hitting milestones and a score (dungeon score).

Dungeons has 5 classes, Healer, Mage, Beserker, Archer and Tank each class has it's own purpose on each floor, try them all at the start and find which one you find the most enjoyable. Later floors particularly mastermode do have a meta that help make things easier but getting started just find something you enjoy and work from that.
wiki.hypixel.net/Dungeon

Using combat minions for combat XP isn't recommended, they're mainly used to get the collections unlocked for each mob. But if you want some extra combat XP then Minions aren't too bad. Just chose a minion that you need to unlock a collection for and you'll get the combat XP as an extra.
wiki.hypixel.net/Minion#Combat_