This is by far one of the easier missions in the campaign. You're objective is to reach the Protoss artifact before the Zerg do. The north part of the map is primarily off limits as its where the Protoss will be battling the Zerg, think of it as a progress bar, the closer the Zerg get to the Artifact the closer you are to losing. Honestly though, on normal difficulty the Zerg move at a snail’s pace trying to get through the Protoss defenses, and there's little chance of losing to them.
1. Assemble your crew
Immediately after starting Swann will send you some Marauders, these are heavy marines with rocket launchers that do big damage to armored targets but have a slower attack rate. Team them up with the marines and medic you started with and make quick work of the Stalkers.
2. Collect the first Relic
You can see the first Protoss Relic from your base but you'll have to take a detour through a small Zerg and then a small Protoss outpost. Be careful as you can lose troops if you're not careful. Once that's collected, loop back around to your base and continue building units. You don't need a large force to overwhelm the Protoss; aim for 4-6 Marauders, 2 Medics and 6-7 Marines. Feel free to use the Mercenary units to speed up the process. You can build an additional Barracks if you're really impatient.
3. Begin your attack
When attacking the Protoss, take out the Nexuses first if possible, as the Proton Cannon turrets will shut down once the nexus is destroyed. You'll encounter some Sentry units while crossing the bridge that will use their barrier move that prevents you from crossing. Blow up the Sentry units first as they'll create barriers and shields that will protect their units. Hang back to avoid taking damage from the Stalkers and wait for the barrier to drop before pushing forward.
4. Collect the second Relic
The second relic is south after crossing the bridge. Continue clearing out the Protoss. If you're in a hurry don't worry about razing all their structures.
5. Collect the third Relic
The third relic is in the South East right before the base. Once you've reached the base, take out the enemy units and destroy their production buildings so they can't attack you from behind.
6. Collect the final Relic and secure the Artifact
The final relic is to the east of the Protoss base, down a narrow path with Nexuses on it.
Once you reach the Artifact 3 large statues will come to life and attack you with strong melee. Focus fire each of them and they'll go down quickly. If you're looking for the rock solid achievement bring additional medics and some more firepower to ensure you won't lose any units.