Mining can be profitable; and certainly seems more fun than fishing, to me.
Here's a quick primer guide, by yours truly.
Mining and You; the Path of the Pickaxe
It is arguable that the entire Republic of Bastok was founded around their mines, and in accordance, most acceptable locations to mine are found in the vicinity of this nation.
The earliest experience most people have with mining is in the Zeruhn Mines, a relatively safe, low-level area just off of Bastok's Mines District; in fact it is involved in Bastokers' very first Mission. The monsters in this area consist of Tunnel Worms, Ding Bats, Mouse Bats, and the single aggressive monster, a Giant Amoeba (~lvl 6).
While the location for the Zeruhn Mines is convenient, the yields are not the greatest. Most commonly mined here is Copper, Tin, Zinc, Iron, and on rare occasion Darksteel Ore, along with Pebbles. The Mining Points appear anywhere in or below the main room (where Smoke Breath and Subodh can be found). There are three tunnels, a short one to the east, and two longer ones to the SE & SW, extending from the main room.
The Palborough Mines are usually the next step up; and can be a greater challenge to mine. There are Mining Points on all three levels, and a character below level 25 will find the aggressive monsters to be a problem. However, the rewards can potentially be much greater here, as in addition to the above ores, Palborough has veins of Mythril & Platinum.
There are at least two other locations where mining can be performed, but doing so below higher levels is very hazardous. The high-end ores available include Colored Stones, Gold Ore, and Orichalcum Ore (heh).
There is a random element to mining; some trips will be good, others you may in fact take a loss. In my experience, even one piece of Field Gear helps your returns while mining; though the armor does require quite a bit of capital gil.
When Mining, know that as you mine out one Point, it will respawn in another location -- you need to keep moving to find the newest Mining Points. As you gain experience Mining, you will also notice that while there are a lot of potential locations for Mining Points to appear, they are fixed and finite.
You can purchase pickaxes in either the Port District at the General Goods store, or in the Mines District from Boytz's Knicknacks.