Taugenichts, that is GREAT. Starting at the gym is one of those things you will thank yourself for later down the road.
I'd say give it two weeks, and the first thing you will notice is an increasing emotional "afterglow" after your workout. You'll walk out of the gym, and feel a sense of natural high, a sense of accomplishment, and a sense of being contented. Regular exercise increases hormone levels in the blood stream that contribute to an overall sense of well-being, and a sense of "hardiness", or being able to face challenges in life and face them down.
At around that same time frame you will start to notice an improved quality of sleep. In addition to the restful properties of sleep, you will awake feeling more refreshed, and feeling like you slept "deeper" than you normally did.
Soon after that, you'll start to notice that you have better energy levels throughout the day, and that you won't have the typical late-afternoon "crash and burn" of post-lunch sleepiness. You'll be more productive and keep a clearer mind throughout the day, and have more consistent energy.
Muscular and physical improvements take a bit longer, I'm not sure what kind of shape you are in now, but you should start seeing some modest improvements or signs of improvement within a month. Don't get disheartened by slow progress.... these improvements occur at the same pace as a huge boat being turned around 180 degrees.. they start slow, but the pace of improvement picks up quickly.
Working out is a great habit to pick up. Even though I have been off and on with it, I think that the overall efforts I made in my younger years to go to the gym, swim, and keep in shape still pay dividends now in my 30's as I still have a very nice physique, I am still lean, and I can still go out and play football with the kids whereas a lot of my friends who didn't do much for themselves physically in their younger days are now more inflexible and feeling more aches and pains.
All the best! As for advice, don't worry about supplements or any of that crap. Just concentrate on consistency. Instead of going to the gym twice a week for two hours, a better strategy is to go four or five times a week for just 45 minutes to an hour. I think an hour is perfect, 30 minutes of weights (for muscle growth, bone strength) and 30 minutes of cardio for the heart and overall metabolism (start slow, perhaps doing 30 minutes of fast-paced walking, then slowly increase the speed until you are doing a mild jog pace). A radio walkman/ipod, or music in general helps with this. I program my ipod every week with music that will inspire me as I run!.. it's kind of fun, I have 5 minutes of warmup music, and then I have the theme from Rocky programmed into the ipod, signalling me that it is time to turn up the heat and start the full-speed run. Be creative, and have fun with it.