I flew RC planes for 10 years until a variety of factors convinced me to leave the hobby. It was getting too expensive for one, planes required constant maintenance, and the club I belonged too started getting very unfriendly. I do the cars now. Just lost the passion for flying.
Anyway...
You'll want to start with a nitro or electric trainer plane. Wing mounted high with a flat bottomed airfoil for stability. Electric can have a higher up front cost but its cleaner than nitro and requires less maintenance. I can't really recommend a trainer as I don't know what's the best on the market anymore. If you're a beginner, make sure you go to a decent local hobby shop with a knowledgeable staff. They'll be able to point you in the right direction. I do highly recommend going the electric route though for beginners. So much easier.
For electric, you will need...
-A trainer plane (duh), Bigger planes tend to be more stable. Don't be tempted to buy anything other than a trainer as they are too difficult for beginners and you will crash it. Get one that already has servos installed.
-LiPo batteries, probably 3S. Capacity depends on size of the plane. You'll want at least two packs. Make sure they have balance leads. I get them from HobbyKing. They come from China but they're not bad quality. A lot cheaper than here. Markup on batteries is ridiculous at North American hobby shops.
-LiPo charger and Balancer. I use the Electrifly Triton Jr though there's probably newer ones. Some chargers have a balancer built in. If it doesn't you can get external ones. They go between the charger and battery, and make sure each cell is charged to the same voltage. Prevents premature wear.
-2.4ghz radio. Some planes come with them as part of the kit. Make sure it's 2.4ghz. The only 72mhz is obsolete. They still sell them but they're not safe to use anymore IMO. If you buy separately, most of the major brands (Futaba, JR, Spektrum, Hitec) are all good. Get at least a 4ch. Though if you get a 6ch, it will be good for 99% of what you will do in the future if you continue in the hobby. Most radios come with rechargable battery packs. If they don't and it just takes AAs, just buy some rechargable ones from Walmart or something.
-12v 5A DC bench power supply. One that has terminals for banana plugs and alligator clips. Most electronic supply stores sell these. Very handy for charging batteries as a lot of field chargers are DC only.
-Tools. You'll need a set of screw drivers (Phillips) hex drivers (I recommend one of
these), needle nose players, blue lock-tight, socket set, x-acto knife, CA glue, epoxy, and a soldering iron.
That's all I can think of for now.
For nitro, you will need...
-Trainer plane. .40 ones. I always used to like the Hobbico ones. If they still make the Avistar, I always felt that was the best. Most ARF (almost ready to fly) trainers come with a two stroke engine and the appropriate propeller. If not, the hobby shop staff can tell you what you need. They also usually come with their own servos and battery if you don't want to screw around with building.
-2.4ghz radio, as above
-Glow fuel. 10% nitro is good enough for beginners. Comes in pink or green. Only difference is oil type. Pink is castor oil, green is synthetic. I've used both oils and I can't tell the difference.
-Glow ignition. Looks like a metal stick with battery on the top. Heats the plug.
-Spare glow plug, look for #8 or A3 plugs. I like the hotter A3 myself but #8 works well.
-Electric glow starter. Turns the prop with a big ol' electric starter. Safer than the old chicken stick method IMO.
-12V utility battery to power the starter motor.
-Anchor to hold the plane when running up the engine. Big screw driver (use it as a stake in the ground) and a bungie cord will do.
-Tools. Same as you'd have for electric
-Windex and paper towel for cleaning your model at the end of the day. Nitro engines don't burn the oil so planes get quite slimy after a few flights.
Once again, I think that's all.
Next step is to get an AMA membership if you're in the US. It provides liability insurance if you crash and injure someone or damage property. Then you need to join a club with a decent training program. Experienced pilots will check over your plane and teach you the basics of how to fly safely and how to maintain your model. Going through a good training program is key to having a good experience. If you go out on you're own, you're guaranteed to crash.