At one point Anthem's campaign puts everything on hold and forces you to grind out four challenges in Free Play to move the story along. It's annoying, but you can blast through the challenges quickly and race on to much more exciting missions.
Here's a breakdown of the objectives you need to complete and some advice on how to tick them off quickly. Note that you'll mostly want to do these solo because annoyingly the game doesn't share credit for kills and treasure chests among the group at the moment, though this is likely to change in future patches.
The quest is simple, just a bit arduous. Complete all the challenges for a tomb, and the tomb door will open. Once all the tomb doors are open, the main story can continue.
Tomb of Gawnes Challenges
Location: The uppermost tomb to the west. This tomb is behind a waterfall in a swampy area infested with Skorpions.
Melee defeats x 50
'Defeats' implies that you have to lose to melee attacks here, but actually it's about enemy kills.
Fly around in Free Play and dive into any world events that pop up. Raid the Outlaw camps to the east of the map (Outlaws tend to spawn in droves) and avoid encounters with large individual Titans.
Use the Interceptor Javelin, if you have it unlocked, and make use of its spinning diving melee attacks from midair to deal lots of of damage.
Ultimate defeats x 50
Kill enemies with your ultimate. The Interceptor is good Javelin for this again, because its super melee state lasts ages and lets you chase down scattered enemies.
The Colossus' big area of effect blasts are great against clustered foes, so again look for Outlaws or Skorpion nests.
If you see a chest out in the open there's a chance it's a Skorpion ambush. These are good for getting lots of kills because bugs have to rush into you.
Legendary defeats x 3
Kill three Legendary level characters. These enemies have yellow health bars and 'Legendary' in their names. They tend to spawn at the end of public events, or as big bosses like Titans in the open world.
Tomb of Cariff Challenges
Location: Underneath a huge fort in the northernmost point of the map. There's a basement entrance to the front of the fort, follow the path around to the back and you'll spot the tomb easily.
Missions x 3
Complete three expeditions for folks in Tarsis. World events should count as well.
Gear defeats x 30
Get kills with your Q or E abilities. Area of effect attacks like the Colossus' flamethrower are excellent for this. Equip it and find some Outlaws or Skorpions to roast. Storm's pulsing fire AoE attack also completely melts hordes.
Combo triggers x 15
Primer abilities inflict a status effect on enemies, and you'll recognise them by the little circle by the ability name and HUD icon. Once the status has taken effect you'll see a red icon above affected enemies. Then hit them with a detonator ability: anything with a four-pointed star next to the name.
All Javelins have the ability to set up and detonate their own combos. Here are some good go-to examples.
Ranger: Frost or Inferno grenades to freeze or burn, then Seeking Missile to detonate.
Interceptor: Venom Bomb or Venom Spray breath to inflict acid damage, melee to detonate.
Colossus: Flamethrower to ignite, Lightning Coil to detonate.
Storm: Ice Storm to freeze, Glacial Spear to detonate.
Multi-kills x 3
BioWare defines a multi-kill as "8 kills with a 10 second timeout between each" and it doesn't matter how you do it. Again, setting off the combos above against hordes of weak Outlaws and Skorpions will get you multi-kills very quickly.
Tomb of Yvenia Challenges
Tomb location: This tomb to the south west is a short flight to the left from the launch platform. Fly low and spot a river running south into a cave (pictured above). The tomb is to the right a short flight down this tunnel.
Treasure chests x 15
The worst one. Find chests in the open world, open them at the end of World Events, and during Strongholds. Our guide to finding chests in Anthem will help enormously here.
You have to be the one to open the chest. Right now if someone else in your team does it, you won't get the credit but this is due to be fixed in a patch shortly.
Harvests x 25
Press F on resource points in the open world. They can be minerals or plants, anything that you can press loot that isn't a chest. Do this in Free Play where the game won't try to pull you ahead to the leading party member.
Javelins repaired x 3
The one objective where you'll want to team up. Get into a group and find some turrets or a world boss, then revive each other until everyone's achieved this objective. If you want to do it solo it then run the Stronghold a bunch of times. Javelins actually go down during this activity, even on Normal difficulty.
Collectibles x 10
Notes, intel, hidden messages should contribute towards this total. Fly up to any tower top or odd piece of terrain and you will probably be rewarded with a note or a scroll for your Cortex. There are a lot of hidden messages and notes in the caves around the Tomb of Yvenia.
Tomb of Artinia Challenges
Tomb location: Ignore the enemy base and head to the stone pillared structure at ground level where the map marker is.
World Events x 5
Cruise around in Free Play until a purple icon appears. They only take five or ten minutes to complete and you can solo them easily on Normal difficulty. Use them to tick off combat objectives for other tombs and claim the chest at the end for the Yvenia chests objective.
Weapon Defeats x 30
Shoot 30 enemies with guns until they are dead.
Weak Point Defeats
Shoot 30 enemies in the head with guns until they are dead.
Elite defeats x 9
You run into these more frequently than Legendary enemies. They have yellow health bars and 'Elite' in the name. They turn up in World Events, but you can also find them milling around in mobs as you're flying around.
And there we have it. Phew! Now onto some spectacular story missions and, in good time, the endgame.