If you are going to have just one plane, you should have a jack plane. It can do the job of the other planes. It is most commonly used to thickness and flatten boards, but it can also joint and smooth boards.
Thickness and Flatten
Despite the fact that they often look straight and flat in the lumber yard, boards are rarely as flat as you need. They cup, bow and twist when the humidity changes. So you need a jack plane to get them flat again. While flattening you can also take the opportunity to remove some of the heft of the board and get it close to its final thickness.
For flattening, the mouth of the plane is set wide open, and the iron is ground to have a curved cutting edge instead of a straight edge. This allows the blade to scoop out wood as it cuts.
The first step in flattening is to figure out where the board isn't actually flat. If the board is twisted, then you'll need to focus more of the plane's cutting on the corners that are twisted up. If the board's face is cupped, then you want to remove more wood from the edges. If the face is bowed, then you want to remove more from the middle. Once you've established where most of your flattening work needs to be done, you can use a pencil or piece of chalk to scribble on the board in the areas to focus on. The nice thing about planing is that you're removing the top layers of wood, so scribble away.
The typical image we have of someone using a hand plane is of them running it along the grain and slicing out a long wispy curl of wood. It turns out that this is not the only way you can do it, and long wispy curls don't do much to make a board flat. So flattening a board starts with the plane moving 90 degrees to the direction of the grain. It is amazing how quickly and how much wood this technique removes. Don't worry too much if your plane is leaving visible tracks, the primary focus is to get things flat. The subsequent steps exist to remove the tracks.
Once you've covered your board with lovely cross-grain planing marks and have a board that is quite a bit flatter, it's time to change directions. Start by planing in a direction that is 45 degrees to the direction of the grain. Instead of short clumps of wood shavings, this will give curly shavings, and the board will start to get smoother. At this point your board will be noticeably flatter. As you're working diagonally, you might want to retract the blade some and take shallower cuts.
Once the board is covered in diagonal tracks, change directions again. This time go 45 degrees from the grain in the opposite direction (that is 90 degrees from your existing tracks). Again you may want to retract the blade some more. With each step the track marks will be shallower and the board flatter.
The last step in flattening is to plane in the direction of the grain (like you imaged you would be all along). Again you probably want to retract the blade some more. Plane until the board is flat and smooth.
The real last step is to check for flatness again, and if it is flat you're done. If it isn't flat, grab your chalk and go back to step one, but you don't have to start out quite so aggressively.
All in all, this process goes a lot faster than you might expect. The cross-grain work at the beginning removed a lot of wood quickly, because it exploits natural weaknesses of the wood. The diagonal cuts tend to mostly remove the weak ridges left behind by previous steps. The straight cuts will remove the diagonal tracks quickly because of the same cross-grain weakness of the wood in the ridges. Now you're ready to flip the board over and do the same to the other size!
Jointing is the process of making the edge of a board smooth and square to the face. A jointer plane is typically much longer than a jack plane, but the jack plane is long enough to joint smaller jobs without any problem. It can be used on larger jobs in a pinch. For jointing, you want the mouth to be fairly tight. Some people like to use a curved blade and others like a straight blade. Most of the time boards, are thin enough that it makes little difference.
Smoothing what old-school woodworkers did before sand paper became available and popular. Smoothing planes are smaller, have a tight mouth, and a straight blade (which might have a very slight curve at the edges to prevent tracks). The jack plane can easily be configured such that it is a large smoothing plane. Planing goes along the face with the blade extending the smallest amount that gives an extremely thin shaving.
The Jack of all Trades
This plane gets its name from the phrase that dates back to at least the early 1600s, because it can do the job of three planes quite well. Some people purchase a second blade so they can easily switch between tasks. Having planes optimized for specific tasks can be great, but if you are getting started or want to invest in a set of planes, you will not go wrong with starting out with a single jack plane.
How do I get one?
There are two main options when buying a hand plane. You can buy vintage or buy new. Stanley made some great hand planes up until WW2. You can find them on ebay, at flea markets, at yard sales, at estate sales and at tool meet ups. You can get some great deals, though you should expect to put in a lot of work to get them into good working order. If you go the new route, then you'll pay more up front and pay less in time later.
In the model numbering system that Stanley introduced over a century ago, you want to look for a #5 bench plane. They all also make a #5-1/2, which is wider and heavier. Both planes work great. Which you choose is entirely up to you. Heavier means you might get tired more quickly, but you might like the weight and wider blade.
Here are a couple options from companies whose planes I've purchased in the past.
Lie Nielsen - $245 - Low angle jack plane. Lie Nielsen is the premiere tool company. You will pay more, but get tools built at the highest possible level around. Made in the USA. The cool thing about this plane is that Lie Nielsen sells extra blades pre-ground for use in different planing tasks.
WoodRiver - $195 - #5-1/2 jack plane. WoodRiver makes a line of very high quality planes modeled after the Stanley Bedrock series of planes. I own this plane and it is one of my favorites.
Woodriver - $170 - #5 bench plane. This is the narrower, lighter version of the plane above.