Given the description of the rest of your body, my suspicion is that the "double chin" is loose skin hanging off the bone rather than excess fat. Of course, I can't be sure without actually seeing it.
As far as exercise goes, if the equipment available is versatile, you don't need a lot of machines. Most of my training over the last 3 years has just used an adjustable barbell, a bench set-up, a cable machine and a power rack. That's 4 main tools, which most self-respecting gyms will have. These are all versatile pieces of equipment that allow for a large number of exercises -- more than you'll ever need to worry about.
There are 6 basic movements that most people should have in their programs:
- Squats/deep knee bends: performed with bodyweight, barbells, dumbbells, kettlebells and machines (such as the leg press), and with various stances and postures, squatting exercises are good for the quadriceps (front of thigh) and glutes (bum). Depending on how they're performed, they can also be good for the adductors (inner thigh), hamstrings (back of thigh), and the musculature around the trunk (back and abs).
- Deadlifts/hinges: these develop the glutes, hamstrings and lower back. Depending on how they're performed, they can also be good for the lats (musles which run from the shoulder down to your lower back), traps (muscles between your shoulder blades), quadriceps and grip muscles (which are in the hands and forearms).
- Bench Press/horizontal push: these movements develop the pecs (chest), anterior deltoids (front of the shoulder) and triceps (back of the arm). This movement includes bench press variations, dips and push ups.
- Overhead Press/vertical push: these movements develop the anterior and middle deltoids (front and side of the shoulder), upper pecs, traps and triceps.
- Pull Ups/vertical pull: these movements develop the lats, lower traps, posterior deltoid (back of shoulder), biceps (front of arm) and grip muscles. This movement includes pull ups and chin ups (with assistance, bodyweight or extra weight added) and lat pull downs.
- Rows/horizontal pull: these movements develop the traps, posterior deltoid, lats, lower back, biceps and grip muscles.
With these 6 movements, you develop most of the musculature of the body. The only other things you're likely to benefit from adding early on in your training life would be calf raises (for the lower leg) and some direct ab work. If you can include some form of each of these 6 movements in your program, and you work on good technique for each movement (as a general rule across the majority of exercises, good technique almost always includes a "chest up" posture and a firm contraction of the abs, although there will be plenty of additional tips for each individual exercise), good things will probably happen.
Other than that, don't be afraid to eat. Your main concern off the bat was that you're too skinny and weak. Food makes your body bigger. Food in the presence of strength training makes your body the right kind of bigger (ie you get lean instead of fat). Eat enough nutritious food to gain 1kg/month, and you'll gradually build strength and fill out your frame without doing too much fattening up in the process. If you're actually underfat (yes, there's such a thing as having too little bodyfat, for either health, function or aesthetics), you might even want to go for 2kg/month to build some muscle and put a bit of smooth padding on top.
