If I want to repair newer laptop motherboards including SMD, BGA, QFN and all of that, can I jut use any iron or does it have to be a special micro soldering iron? Also, will every tip fit every iron?
You really want hot air for this. What hot air does is let you heat the entire component at once, so that if properly distributed solder and flux is present, surface tension will let the chip "float" into aligned position. It is also about the only worth-the-trouble way to remove a bad or misaligned part.
And that, or an oven, is about the only thing you can even try for a BGA. Many would tell you that manual BGA rework without sophisticated temperature profile tools and optical alignment jigs is impossible; in practice if the board would be considered scrap absent your effort, they can sometimes be fixed that way - recently had to rescue a prototype board I was sent to bring up which turned out to have BGA MCU canted at an angle and the balls looking like lava flow fragments.
I don't have a lot of money to purchase equipment or even tutorials however, so please don't recommend any tool over $50.
Well, there are desires and there is reality - fortunately they only differ by moderate pain here. An inexpensive combination hot-air/iron station can be had in the $60-70 range. Maybe less if you order it from overseas on the slow boat, but make sure you get the one with your country's input voltage range. And do expect to have to buy some quality tips.
can I jut use any iron or does it have to be a special micro soldering iron?
You will of course want a soldering iron too - for attaching wires and larger components, and if you want to place a leaded surface mount part by tacking first one corner, rechecking alignment, tacking the other, etc.
One of the keys to using an iron is to remember that it is flux and surface tension, not a tiny tip, which manages most tiny connections. With something like a TQFP, once you have it in position, you can actually just use a big tip to wipe solder down the whole row of pins. Everyone has their preferences, but while this tip looked nothing like I expected from a written description, it has turned out to be what I use 90% of the time for everything from power lugs to tiny surface mount.
The one thing this is not so great for is trying to get at the angle between a pad and the metalization on the side of a QFN IC. Sometimes this will not wet, and being able to get a small iron tip into the corner and rub it a tiny bit does the trick. Another approach is to heat the area with the fine hot air nozzle and poke at it with a sewing pin held in needle nose pliers - but you want to do this under a binocular microscope, so that you see when you succeed in forming a fillet, and don't bump the entire IC out of position.
When you do make a mistake and bridge a tiny pin or two, the first thing to do is to add more flux, wipe your iron, and and see if the iron will reclaim enough of the solder to clean the bridge. For the cases where that does not work, you reach for your roll of the narrowest diameter desoldering braid they sell.
Also, will every tip fit every iron?
No. But clones of a given design, like the Hakko will probably fit the original, and each other. Quality of clone tips, particularly the plating and fit on the heating element varies a bit.
Also, the cheaper solutions often do not do well with tiny tips - it can be very hard for an iron to pump enough heat through these, for example if you have a QFN pin tied to an internal ground plane that soaks up heat. The Metcal and similar RF-heated irons would probably be worth the money if you were doing a lot of such rework... but you'll spend a lot there, even for a tip.
The one thing you haven't asked about is inspection. You really need to be able to see details finer than you can with the unaided eye - even in your teens ;-)
The budget approach is a 10x eye loupe - this will cost you $5-10. Once you have done some work, if you hold this to your eye and get within a couple of centimeters of the board, you can see the results. But the working distance is far to short to try to use it to see what you are doing, especially with tools that are hot and/or sharp.
The usual professional bench tool is a "Stereo Zoom" Microscope. These start at about $300 for nice import models, and easily run to five to ten times that price for traditional name brands, but it's not really clear what useful capability you are getting for the premium name. If you do make such an investment, the combination of a .7 to 4.5x objective, 10x eyepieces and a half power Barlow lens works very well - the latter as it doubles the working distance and puts your zoom into a more useful range of around 3.5-20x (about the only time I have been tempted to remove that to get full power was for peaking under the edges of a BGA). If you only plan to work on small boards, the usually biology-lab base is fine (just replace the plastic circle with a surface that can take the heat) but if you expect to work on larger boards or entire laptops then an overarm model might work.
There are also video systems - historically the USB ones had problems with time lag, though they've apparently gotten better, and today prices range down to $20, so that might be something to try. Often the stock holders are poor, but that is something you could come up with a replacement for. If you need to take pictures to document something, they may actually work better than a traditional microscope - it is hard to get an affordable camera attachment that captures the same field of view as a microscope eyepiece anyway, since your eye is larger than a consumer-gear image sensor.