I used one of these for a very long time until I got a Weller: Velleman Low-Cost Soldering Station
It was pretty decent and was also about $20 USD. I only ever really turned it up for working with Lead-Free, which I tried to use on anything I was building new as a best practice. It only really has the pencil tip available, but that was still better than some of the other irons I had used before.
I'm going to echo what some of the more experienced modders and fabricators in this thread have said: Start with a 15W or so iron. You want to develop good habits when you are learning to solder. You don't want to destroy vintage equipment. Since we've all done this a fair amount, we definitely have grown to like repairing and making electronics, so once you get "the itch" and you have a good technical understanding of the difference between crappy soldering (cold/dry joints, loose tin, etc) and quality work that's going to last and keep your toys ticking, then it's time to move up to a 50 or more variable/temp controlled iron.
Welcome to the club!
P.S. Any other quick tips I had to pass on are summed up like this: The soldering iron is just one tool in a workstation that enables your to make what you want to make. Even with a decent (or a great) iron, if you can't keep all of your stuff in place while you try to apply heat and tin, you're going to end up frustrated and with a sub-par end product. I think having a good mini vise / third hard is at least as important as the quality and wattage of whatever you're using to tin and solder things together.