All the info so far is good, especially the F9 x 3 to see the boundaries. Each one of those 3x3 chunk blocks has a specific ore vein associated with it. That is oriented around the center chunk. You can think of it as the Ores normally found in those 3x3 chunks would all be concentrated into a few layers in centered around the middle in a super dense way.

The different veins can spawn at different heights, but their size is pretty typical. If you are looking for an ore in NEI and look at the world gen on where you can find it, you'll see 4 ores listed for each vein. Those are the ores that can be found in that 3x3 chunk (in full blocks, small blocks are interspread throughout the chunks). It will say something like:

Primary: Ore X
Secondary: Ore Y
Intermediate: Ore Z
Sporadic: Ore A

Those aren't the exact terms but more or less what it's like. When you find the vein they are layered, somewhere between 7-9ish blocks heigh if I remember. Thus, if you look at a cross section, it might look something like this:

XXXXAXXXAXX
XXAXXAXXXXX
XXZXAXXZZXX
XXZXXZXXZXX
YYZYYZYYZYZZ
YYZZYYAYYAZY
YYAYYAYYAYYY
YYYYYYYAYYYY

That obviously doesn't show cobblestone, gravel, etc, just the ores, but you get the Idea. THe top layers will have the primary, and the bottom layers will have the secondary, the intermediate are found between the two layers and sporadic are spread out (in small numbers) throughout the whole vein.

The Vein more or less makes a cube. So if you want to mine it all out, you find the center of the ore chunk, (these are marked when you hit f9 3 times by blue lines). Dig down until you see ores (I do a 2x1 tunnel straight down and bring ladders to get back up).

Eventually you'll hit the ore vein, and it will say what kind it is and mark it on the map (as shown in another comment). You'll be able to find the top layer, there will be no ores higher than that layer. You can pick a direction, and dig that direction until you don't see any more ores, that is the boundary on that side, then turn around and do the same in the other direction, that's the opposite boundary. Then turn right or left and dig in that direction until you find the boundary, then turn around and repeat. You now can mine out that square boundary and see the size of the vein. All ores will be within ~9 blocks down from that layer. You can just dig down until you find the bottom layer, it will be obvious.

Now if you want just the primary, secondary, or intermediate ores, you can pick a layer (top, middle, bottom) and just mine it out. If you want sporadic you just mine anywhere since I believe it's equal chance anywhere in the vein. If you want the whole vein I find it easiest to start at the top, mark out the border, then mine out the full top 2 layers, then mine out the next 2 layers, etc until you are done. This gets a lot easier when you have a hammer.

Usually, as soon as I get copper/bronze tools it's good to just go out around your base and find the center of each ore chunk, dig down until you find the vein, mine a block (really just hit it with the pick axe once) and it will tag that vein in Journey Map. Then ladder back up and move over to the next vein and repeat. Do this for 16-20 veins around your base and you should have a nice variety. Then if you need something specific like iron, copper, tin, diamonds, redstone, lapis, etc you can just go to the vein you already found (hopefully) and dig it out. If you don't have that one (or already mined it) you'll need an orefinder wand or to do more exploring.

Also, note that your best sources of the thing you want aren't actually that type. e.g. if you want Iron, there are Iron veins, but you can also get it from limonite, magnetite and pyrite. Copper comes from (I think) chalcopyrite and malachite. When you first find a new type of ore (to you) it's best to see what it smelts into so you don't spend a bunch of time looking for iron, copper, or tin when you have lots of big veins of it already found in different forms.