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I have been having a lot of problems with this railway. I have tried switching the train stops, but I keep getting a no path error. I looked at tutorials and followed the directions in them, but I can't seem to get the railway to work.

Both of your train stops are on the wrong side of the track. They are always placed on the right hand side of the track where the front of the train should stop. So for the west station the stop should be on the north side of the track and on the east station the stop should be on the south side.

Also, the one to the east is not even on the track at all, it must be placed directly along the side of a track piece, and in that particular spot on the south side of the track.

EDIT: You should also know that a train engine can only drive forward when automated, so if you want it to go back and forth on that small piece of track then you need a second engine facing the opposite direction. Alternatively you can create a loop with your track, allowing you to use a single engine which always drives forward along the loop.

EDIT 2: check out his tutorial video on trains: https://www.youtube.com/watch?v=3ZiL7_vfJtQ, it is very in depth and easy to follow.

For this setup

1. [edit] The left station needs to be on the north side and the right station needs to be on the south side of the track.
2. The East train station isn't even on the track.
3. Signals need an oposite signal directly across from them to make the track a two way track (back and forth).
4. Trains on a schedule will only go in the direction the locomotive faces. To have the train go in both directions you need to add a second locomotive facing the other way.

For distances that short, trains are overkill. Belts will do the job just fine. That said, perhaps now is a good time to learn how trains actually work before you actually need them to bring in resources from far away places. So with that in mind, here's what's up with your train setup:

1. Stations are on the wrong sides: They should be to the right of the locomotive as it would enter the stop.
2. Signals are also on wrong side: Just like train stops, signals are also always on the right, but there's a 2nd issue here. You appear to intend for this bit of rail to be a 2-way system, which means you need signals in pairs one on each side at any point you want to put signals (honestly signals are unneeded until you have more than one train in your system, but it can also be a good idea to learn how signals work now as well).
3. Scheduled trains never travel in reverse, hence in order for a train to start moving in the opposite direction, it's needs another locomotive facing the opposite direction (the setup you'd end up with is commonly abbreviated 1-1-1, which denotes #forward locos-#wagons-#backward locos).

Fix those issues and your trains should start running nicely.

Speaking of signals, I want to say a bit more since it's one of the less intuitive things in the game (at least it was for me). First of all there are two kinds of signals, the more basic regular kind (simply called rail signal) and chain signals. The former works like this:

Regular signals on the same side of the track subdivide that direction of the track into blocks, only one train may occupy a block at a time. (Note that for 2-way rails, you need to make 2-way blocks for trains to move through, hence the need for pairing up signals one on each side, otherwise you have a little piece of 1-way rail in the middle of an otherwise 2-way system, and it will prevent trains from travelling in the 'wrong' direction through it).

Chain signals exist to solve a fairly simple problem: Consider a rail crossing (two separate lines, that cross at 90 degrees, suppose you put regular signals on both sides of each rail line where it crosses so that your trains will negotiate the crossing without running into eachother (actually they won't path at all unless this is done anyways, because the train on one line will see ALL of the other line as part of one of it's blocks, and hence the train on the other line is always occupying that block). Suppose you are running a lot of trains on one of the lines, so many that the block immediately after the crossing is still occupied when another train on the same line wants to enter it. So it has to wait, but it's waiting on the crossing preventing any cross traffic from getting through, NOT COOL! Put a chain signal before the crossing, and now the train can't enter that block unless the one after it is clear. Moreover, multiple chain signals sequentially on the same side will (as their name suggests) 'chain' not allowing a train to pass the first signal until the block after the next regular signal is clear. So in short use chain signals before (and within) junctions and regular signals on the exits (sometimes there's enough space for a train to wait in a large and complex junction, there you can have a regular signal on entry to the long piece and chain on exit side; both otherwise stick to the aforementioned rule of thumb).

Thank you all the help. I finally got the trains working. I didn't belt the iron as it was a relatively small mine and I ate through the iron from it in like 20 minutes while the tracks now connect to the much larger iron mine which you can see on the map.