This mission is fairly straight forward but can be a bit confusing until you understand the Protoss' attack pattern. Your SCVs need to extract gas from the highlighted purple geysers and will be attacked as they do so, it’s up to you to send a squad with them for protection.
1. Crank out some Marines
This mission features some Goliaths you can use, but they're very expensive and frankly aren't worth the required cash to upgrade and repair. The easiest way to do this is, as usual, with Marines. Build a Tech Lab on the existing Barracks and an Engineering Bay to upgrade infantry weapons and armor. Once you've got the cash build another barracks with a reactor. Make sure to build many additional SCVs as you'll need to make up for the fact a few of them will be busy completing mission objectives and not gathering resources.
2. Begin capping the west side Geysers
Once An SCV starts gathering gas, it’s usually about 20 seconds before a small squad of Protoss will show up to attack, usually 2 stalkers 2 flying ships and occasionally a zealot. Any small squad of marines with a medic should be able to handle this assault. If you decide to cap 2 geysers at once, it's not a huge problem, just be prepared for a heavier assault. Anything more than 2 geysers at once is not advised.
Instead of capping the nearest geyser, instead tap two of the geysers north west of your base. If you do these first, you can usually get away with only a very light assault on your SCVs that the Goliaths and a few marines can handle. Cap the third geyser in that area and then the fourth immediately to the left of your base. That will leave three more.
3. (Achievement) Collect the Protoss relics
Protoss Relic #1
Protoss Relic #2
Obtaining these two will require you to fight far more troops than you would normally, as you'll be forced to clear both areas. If you choose to grab both, make sure you have a very large group of 20 something Marines to push through the Protoss lines. These relics can be a pain on harder difficulty modes so don't expect to simply waltz in.
Protoss Relic #3
4. Begin capturing the East side Geysers
From this point you can choose to push to the geyser further into the Northwest or Further into the North east. The North west area is recommended as it only has a few small outposts to deal with as opposed to the Protoss base to the NE.
The mission's biggest problem comes from trying to collect all the relics as there is one on each side of the map. the first one is conveniently located near your base but the other two are on opposite ends of the map:
The Protoss will sporadically launch attacks on your base during this mission, but it’s nothing a few marines can't handle. If things get out of hand make sure to utilize the Mercenaries at your disposal. Other than that, this is a straight forward mission that, as long as you don't get greedy and stay patient, is easy to manage.