In Temperate or Subarctic climates, delivering coal to power plants is an excellent start to your budding trade empire. Coal also has no late delivery payment reduction, so you don't need to worry too much about getting it there in a timely manner. You can find a power plant all the way across the map, and as long as you can afford laying the track out there, delivering the coal there will pay a good amount. If you find another coal mine along the way, you can build a Feeder Bay, drop the coal off there, and let a train from this mine deliver the coal the rest of the way.
If you're not in a Temperate or Subarctic climate, take a look at these industry flowcharts. You want to choose something simple, a single item deliver that has no output, if at all possible, to start with.
Further, Power Plants never shut down, so you don't have to worry about suddenly having commodities available that you can't deliver. The coal may run out, but that's a pretty simple logistical problem; just deliver the rest, and then sell the train(s).
Now, it will take some patience, but what you can do is focus on a single route at a time, and ensure you're getting maximum profit from it. If you have a train on a single track that always takes a full load from the mine, and there's some leftover, add more cars to it until it can pick up ~95% of its capacity the instant it gets there. That will lower the amount of wait time to fill it up completely, and ensure the amount waiting doesn't decay too much.
If you have a super long train already, and you still can't carry most of the coal, then it's time to twin the track. Either two side-by-side, each with its own train, or a each side a one-way, so the trains go in a big circle. Turning it into a big loop also ensures you can place more than two trains on the same track.
Once you've tweaked your first route to be as efficient as possible, you can either go create another one just like it, or you can start building your fledgling trade network.
Once you get a decent amount of cash in the bank, you can start building sections of a multi-part industries, such as Iron Ore --> Steel --> Goods. Start on one section at a time to avoid bankrupting yourself, and start at the beginning. First deliver iron ore to a Steel Mill, and follow the tips above. Once you've got that straightened out, start delivering steel to a Factory. The hardest part of doing this is figuring out how to manage the trains that are delivering and picking up goods.
You can build trains that go along the entire route, but they won't be very efficient. It's better to build dedicated trains for each type of good, and let them handle their own sections. You can see some pretty good examples of how to build advanced stations from the OpenTTD Wiki.
