I've heard/read here and elsewhere that I shouldn't be doing cardio if my ultimate goal is to gain weight.
Some limited cardio can be good while weight training. Cardio helps with increasing O2 circulation, strengthens your heart, and increases endurance. However, what cardio doesn't do much of is put stress on your muscles to initiate growth. Running 5 miles a day isn't the same as doing squats because your muscles quickly acclimate and no further challenge is presented.
Cardio is a great warm-up technique for me, personally. I run about about 20 minutes before moving onto the actual weight training. Just enough to warm up most of the muscles in my body, and I've found that it increases my focus better than stretches - which is very beneficial when working with weights. However, if you have a personal trainer (or will soon), they'll be better able to tailor a workout routine for you.
I have a just a little bit of a "beer belly".
My conditioning seems poor. Specifically when it comes to certain nighttime recreational activities where I can't finish before I finish. Should I be running the treadmill or cycling at the gym? Should I do morning jogs, or any cardio at all?
The beer belly will improve with just about any exercise that you do consistently, but things to keep in mind are that genetics may be operating against you (men generally store extra fat on their torso, and some men moreso than others) and that a flat stomach is more an issue of bodyfat % than strong core muscles. While strong abdominals will certainly be larger and push against the underside of your skin more than untrained abs, you won't see a six pack without bringing bodyfat % down. SOME body fat is good. If you ever get sick and can't eat a lot, or simply can't eat for other reasons - your body will preferentially use fat instead of muscle if it's available.
As for your nighttime activities - again, just about any exercise is going to help you. However, squats and some light cardio would probably help the best for that particular activity. Running empty because of muscle fatigue (your heartrate is fine but your muscles abandon you) might push me towards a more strength-training oriented workout. Running empty because your heartrate shoots up and you're breathing very heavily would push me towards recommending more cardio.
As for what you should be doing: Well, that depends on where you are right now, and what you ultimately want. Find an exercise you enjoy and do it consistently is the first step - sporadic workouts don't help, and may end up harming you in the long run. Make a conservative goal; strength train for 30 minutes or jog for 30 minutes. Once you've reached that goal, take stalk of your abilities and re-assess your plan. Once you're happy with one type of exercise - say you're content with jogging for 30 minutes three times a week - then you can work on strength training for muscle mass (or vice versa). After you're happy with both, it's an issue of maintenance.
There are important things to remember, though:
-PROPER FORM: Learn it early, do it often. Cardio injuries aren't acute - they sneak up on you after months or years of bad training if you don't have proper form. Strength training injuries can be acute, and very, very painful. Lower back injuries from improper form are common, and if you get to larger weights, can be quite literally crippling in extreme cases. If you cannot sustain proper form throughout any exercise, STOP immediately and try again another day.
-You might make some early muscle mass gains whilst losing weight, but the overall consensus is that you should choose one or the other based on your goals. Gaining muscle mass involves challenging the muscles and eating surplus calories (usually lean protein). Losing weight involves maintaining muscle mass and caloric deficit (usually through both exercise and decreased caloric intake).
-Strength training is not about seeing how quickly you can perform a set of reps, it is about how controlled you can be with the most weight. When you're ready to perform an exercise, take a moment to flex your muscles and brace your body. It's also very common to start with lighter weights first - maybe half your max - so that you can fix any problems in your form when you're at a lower risk for injury.
-Whatever you decide to do, stick to it. That's the most important party of any exercise regime. Any training you do - whether cardio or strength - will be quickly reversed if you can't keep doing it. Strength gains can disappear in as little as two weeks (depending on how long you've trained the muscle), and endurance gains can disappear in a similar time frame if you take long breaks. Aim for 6 to 10 weeks of very consistent workouts; after that it's usually turned into a (health) habit with all the benefits that goes along with exercising on a consistent basis.
-Rest Days are very important, and you shouldn't force muscles to do what they don't want to do. If you or a particular set of muscles are sore, let them be sore until they aren't anymore. If you feel like you've pulled a muscle, stretched a tendon too much, or have injured a joint - always heal before returning to a workout; even if it means weeks of not training. Don't 'train through the pain' - the wear and tear will come back to bite you. As you progress through whichever exercise you prefer, you'll find your recovery time will shorten.