Well, I don't see what's wrong with a large Adam's apple for man, but I suppose that if everything around it gets built up, the Adam's apple will appear smaller by proportion. After all your main compound lifts (which should make up the bulk of your program), maybe a couple light sets of some neck resistance exercises would help you out, but you've gotta be careful and make sure you do them safely. It goes without saying that you don't want to mess up your neck. It's worth noting, though, that a swolen Adam's apple is a sign of thyroid problems, just so you know.
Anyway, for your program, you should be training legs, push and pull movements. You don't need a lot of exercises (especially starting out), you just need a few good exercises. With your current equipment, for legs you can do lunges, split squats, goblet squats and front squats. If you get a squat stand or power rack, you'll also be able to do back squats. For push you can do DB and BB bench press and overhead press. For pull you can do deadlifts, Romanian deadlifts, barbell rows, T-Bar rows and 1-arm DB rows. If you train 3 times per week and do 1 leg, 1 push and 1 pull exercise each time, you'll be on the road to improved strength and size.
But your current equipment won't be enough to get you where you want to go longterm. As I said before, a squat stand or a power rack would be a good investment, as would a pull up/dip rack, a couple belts (a weightlifting belt for support on barbell exercises and a dip belt to add weight to bodyweight exercises), weightlifting chalk, weightlifting shoes and of course, more plates to put on the ends of your barbell.
Do a program using the movements I described above. Try and buy a new piece of equipment every month or two, from the list I just rattled off. I'd probably go for more weights and chalk first, then a squat stand or power rack, then fill in everything else as you go.
You'll need to eat to put on weight. To gain 300-500g/week (which is a good target weight increase), you'll need to consume a calorie surplus of 300-500kcal/day. That means that if you need 2,500kcal/day to maintain bodyweight (after factoring in exercise), you'll need 2,800-3,000kcal/day to make decent gains. Gaining weight slower than that makes it harder to measure progress, and it means that you've got less room for error. Gaining weight faster than that usually just means more rapid fat gains, which is obviously not ideal.
Within your daily calories, you'll want to be consuming 1g/lb bodyweight/day in protein. So, at 135lb, you'll be aiming for ~135g protein per day. At 180lb, 180g protein. Simple rule of thumb.
Other than that, get plenty of fruit and vegetables in your diet, and you should be good to go.
