After returning to Factorial Town as part of the story in the present day a colosseum has opened up. A staple of many Digimon games here you'll be able to engage in themed fights. While some unlock over time there are also a number of fights that require the completion of Side-Missions in order to progress.
Digimon Story: Time Stranger - The Great OMNI Carnival
The reward for completing all of the Colosseum matches is from the Side-Quest "The Great OMNI Carnival". This is a quest that you get relatively early, but it will take you finding more matches as you progress through the story in order to unlock all of the fights you need.
By completing this quest, you'll earn the following rewards:
- Anomaly Points x200
- Courage Point V x2
- Immortal Egg x1
For this my honest recommendation would be to come back when you're at least Rank 8 in Agent Level, the only immediate benefit from completing the battles that are available to you are the reward of the Omni Blade digivolution item, but until you're Rank 8 you can't even use it, and why make the fights harder than they have to be?

The list of fights, their combatants, and any strategy recommendations are listed below:
Name: Searching for Nanimon
How To Unlock: Unlocked when you first gain access to Paradise Colosseum
Combatants: Lv 46 Digitamamon (Data) and Lv 46 Botamon (No-Data)
Digitamamon uses X-Revive on Botamon so focus on Digitamamon first
Name: Kings of Big Bad Bugs
How To Unlock: Unlocked when you first gain access to Paradise Colosseum
Combatants: Lv 47 Snimon (Vaccine) Lv 47 funbeemon (Virus) and Lv 47 Tentomon (Vaccine)
Name: Monzaemon and Friends
How To Unlock: Unlocked when you first gain access to Paradise Colosseum
Combatants: Lv 45 SaberLeomon (Data), Lv 45 Monzaemon (Vaccine), Lv 45 Unimon (Vaccine)
Name: Lucky Number 100!
How To Unlock: Unlocked when you first gain access to Paradise Colosseum
Combatants: Lv 48 SkullMeramon (Data)
SkullMeramon will use Triple Turn to make up for being a solo fighter
Name: Trustworthy Truth-Tellers
How To Unlock: Unlocked when you first gain access to Paradise Colosseum
Combatants: Lv 49 Whamon (Vaccine), Lv 49 Puppetmon (Virus), Lv 49 Lillymon (Data)
Name: Circus in the Red: No Misreckoning
How To Unlock: Unlocked when you first gain access to Paradise Colosseum
Combatants: Lv 50 Piedmon (Virus), Lv 50 BanchoLeomon (Vaccine), Lv 50 SkullMammothmon (Vaccine)
Name: The Royal Flush
How To Unlock: Unlocked after completing a previously available collosseum match
Combatants: Lv 51 Sukamon (Virus), Lv 51 PlatinumSukamon (Virus), and Lv 51 Sukamon (Virus)
They'll poison your team, but otherwise are pretty weak

Name: Covert Honey Corps: Royal Commandos
How To Unlock: Unlocked after completing a previously available collosseum match
Combatants: 3x Lv 52 TigerVespamon (Virus)
They'll all use Double Action so focus one at a time to reduce numbers quickly
Name: Imperialdramon: Paladin Mode
How To Unlock: Unlocked after completing a previously available collosseum match
Combatants: Lv 55 Imperialdramon: Paladin Mode (Vaccine)
Imperialdramon: PM is weak to Wind attacks. Once you get them to ½ HP they'll use Triple Attack.
Winning this battle gets you the Omni Blade Digivolution Item that you can use at Agent Rank 8.

Name: The True Twenties
How To Unlock: Unlocks after completing 'The East Road is Best' Side-Mission
Combatants: Lv 53 Grademon (Vaccine)
Weak to Fire and Wind, Strong to Elec and Null
Name: Gatekeepers of Fire & Ice
How To Unlock: Unlocks after completing Cosmic Area story
Combatants: Lv 53 Sorcermon (Vaccine) and LV 53 Wizardmon (Data)
They will take turns healing each other so store up for a massive push by buffing your Digimon and try to wipe one out quickly. Putting one to sleep is also very helpful.
Name: Hey, Kid! You Here for a Fight?!
How To Unlock: Unlocks after getting the quest to return to the Government Building
Combatants: Lv 60 Titamin + SkullBaluchimon (Virus)
Once you get to half health they'll split into a Lv 70 Tsunomon and Lv 70 Dorimon. Tsunomon will still freeze time and unleash a special attack, focus on it first, Dorimon will buff and debuff
Name: Lucemon ⅔
How To Unlock: (Unlocks after completing "The Cycle of Time" and traveling to the In-Between Theater
Combatants: Lv 57 Seraphimon (Vaccine) and Lv 57 Orphanimon (Vaccine)
Weak to dark, resists Fire. Seraphimon will use X-Heal so focus on him first. Orphanimon will mostly debuff you.
Name: Golchidrarim
How To Unlock: Unlocks after completing "The Cycle of Time" and traveling to the In-Between Theater
Combatants: Lv 61 SlashAngemon (Vaccine), Lv 61 ClavisAngemon (Vaccine), Lv 61 MagnaAngemon (Vaccine)
SlashAngemon can make himself reflect Magic Attacks
Name: Terrier Twins and Their Teacher
How To Unlock: Unlocked after completing all previous bouts
Combatants: Lv 62 Lopmon (Data), Lv 62 Cherubimom (good) (Vaccine), Lv 62 Terriermon (Vaccine)
Terriermon and Lopmon will hurl status issues at you, try to take them out quick or put them to sleep and save them for later

Name: Golden Intruders
How To Unlock: Unlock as part of the Final Battle Challengers Side-Quest
Combatants: Lv 52 Pharaohmon (Virus) and Lv 52 Rapidmon (Armor) (Vaccine)
Pretty easy fight if you've done the other
Name: Kings of Pharaohmon
How To Unlock: Unlocks before the final fight of the game
Combatants: Lv 75 KingEtemon (Virus), Lv 75 Pharaohmon (Virus), and Lv 75 PrinceMamemon (Data)
KingEtemon will force you to attack him, focus him first or just use attacks that hit everyone. PrinceMamemon will debuff the team. Pharaohmon will also revive his downed allies
Name: Mighty Mad Man
How To Unlock: Unlocks as part of the 'A Stubborn Fighter?' side-quest
Combatants: Lv 55 Gankoomon (Data)
If you've made it to this stage… it will be an easy fight
Name: Solo Beauty
How To Unlock: Unlocks during the 'A Display of Beautiful Swordplay' that you get in the Royal Knight quest line
Combatants: Lv 55 LordKnightmon (Virus)
Name: Master & Pupil Trio
How To Unlock: Unlocked as part of 'The Forgetful Helpers' Side Quest
Combatants: Lv 60 Kotemon (Data), Lv 60 Zanbamon (Virus), Lv 60 Mustamon (Virus)
Name: Hero from Shambala
How To Unlock: Unlocked in the 'Susanomon the Hero" side quest
Combatants: Lv 70 Susanomon (Vaccine)
Will use Triple Attack at 2/3rds HP
Name: Marsmon, God of DigiBattles
How To Unlock: Beat all other Colosseum battles
Combatants: