You probably should start with a power supply, you'll need one of those for any project you do later. Of course I don't recomend get any near to mains for a beginners project, but you could build a "labbish" PS and power it from a few 9V batteries to start and then maybe a plugpack till you get confident enough to deal with the mains yourself, with an electronic adult watching you if possible. Two LM317, one limiting the current and then another regulating the voltage could be quite useful. The current is quite tricky to select with just a pot but maybe 3 option switch (50mA, 250mA, 1A for example) could save you a lot of ICs. I was saying in some other post burning (cheap) ICs is a good thing but having a suitable PS is too.
Other than that, I started with guitar pedals, a great deal if you are a guitar player, if you are a friend of a guitar player he will be thrilled, and probably will end sucking your life and financing you a lot of fun.
Audiorhythmic lights / VUmeters
Anything you can imagine battery powered as long as you don't want world class quality nor years of battery life is, most likely, beginner friendly. Thanks to Arduino µControllers are quite accesible for beginners and that opens a whole world which wasn't that handy when I started, today I received a bag full of those for tiny dirty projects I have, a thermometer took me 2h to have running and while using it in a weekend I tuned it as much as I liked for example. Records max and min temps since reseted, shows in a display. You don't need much at all for that, the arduino, a display and an LM35 (temp sensor LM34 is easier to use in ºF if you like). I guess you could have it running the first weekend you get to play with it, probably the first day.
Go for it, you will get great things out of this, you will blow some components in the way, don't worry, 14 years later I still keep calling the first power one of a project the smoke test. Just be safe, no mains till you are a grown up. Hi power stuff is fun but I didn't get into it for a few years, better I don't say my age of my first time playing with high voltage tube stuff! FUN
JS