Walking past the dead Midgar Zolom and into the cave will lead you into the Mythril Mine.
There are two important enemies which you should watch for as you travel through the Mythril Mine: Madouge and Ark Dragons. Screenshots of each of them have been included below.
You can steal a weapon for Tifa called the Grand Glove from a Madouge and you can learn the Flame Thrower ability through your Enemy Skill Materia when you encounter an Ark Dragon. The Grand Glove is an excellent upgrade for Tifa and is most certainly worth the effort of tracking down if you plan on using Tifa in your party.
From the entrance, travel to the right side of the screen, around the corner and to the north. You can climb the vines on the left side of the next screen to find a Long Range Materia on the cliffs above. Long Range Materia can be used to allow one of your melee characters (Cloud, Tifa, Aeris or Red XIII) to attack from the back row and receive no reduction in their attack power while still obtaining all of the defensive benefits of being in the back row. Check out the Tips and Tricks section for more information on front and back row mechanics.
Climb back down the vines and continue north in this area of the cave to find an Ether on the ground and a treasure chest that contains a Tent. Return back to the entrance of the cave one you have retrieved all three items.
Travel to the southeast portion of the cave by heading southward away from the entrance and you will find a treasure chest that contains a Mind Source. Head west from here into the next area and you will run into the Turks: Rude, Elena and Tseng. Go north once the cutscene and dialogue has concluded into one last room which contains a Hi-Potion and a treasure chest with an Elixir in it.
Go back to the previous screen where you met up with the Turks, climb the vines and exit the Mythril Mine. You are now in the Junon area and, as mentioned by Elena, Sephiroth was headed to Junon which is where your team is now headed to. There are a few things that you can do prior to going straight for Junon though…