I recommend checking out the stickies for some posts by the member "steve" on energy balance. Basically, though, you can't really lose weight and gain muscle at the same time. Losing weight requires a calorie defecit, while gaining muscle requires an energy surplus. When you lose weight, you lose fat AND other tissues, including muscle. Lifting while losing weight is meant to force your body to preserve as much muscle as possible. When you're trying to gain muscle (i.e. eating in a surplus), you will gain weight, but that weight will include fat.
That being said, if you've never lifted before, or haven't lifted in a very long time, you will be able to get much stronger and possibly gain some muscle even if you're eating in a defecit to lose weight. I started lifting for the first time in years in july - I workout 3 days a week, alternating between these 2 workouts A: Squats 3x5, Bench 3x5, Assisted Pullups/chinups 3xF, B: Squats 3x5, Standing Overhead Press 3x5, Seated Cable row 3xF. I use the same weight for all 3 sets, and each workout day I increase the weight lifted by 5lbs. My full squats have gone up 80lbs since I started, bench has gone up 45lbs, and Overhead press has gone up 35lbs. Takes me only 45mins per workout, 3 times per week. I've also lost 10-20lbs during this time frame. I don't know if I've gained muscle mass per se, but I can tell when I feel my pecs or my arms that my muscles there seem bigger, stronger, and more defined.
Be wary of trainers. They don't always know what they're talking about, and some tend to tell you only what you want to hear so you'll buy the sessions - you're more likely to buy the sessions if they tell you what you want to hear vs. whats correct or best if its not what you wanted to hear.
I wouldn't say its necessary to lift weights twice in the same day, even if its different body parts. As you can see from what I posted above, its possible to make lots of progress quickly without spending hours in the gym every day. I'm not saying you should go out and do my routine exactly as posted, but I would recommend you stick with free weights rather than machines, and do lifts which incorporate compound body presses (squats, bench, deadlifts, overhead press) and pulling movements (pullups/chinups and rows), and ignore stuff like bicep curls and trcep extensions and other excercises that only work small, weak muscles - those muscles will strengthen and grow just fine without isolated work. Don't believe me? Go bench press and follow it with assisted pullups - I guarentee you'll feel the burn in your biceps. I would also recommend higher weights and lower reps if you want to build strength/muscle. Something like 3-5 sets of 5-8 reps seems like a good range to work in. More reps/lower weights is good for muscle endurance (workout longer before fatigue), but you wouldn't be pushing your muscles as hard to encourage your body to adapt to heavier weights by getting stronger/building muscle.
There are some stickies around (also by "steve") that go into much more detail than I have posted. I recommend trying to find some of them.
