Don't be like the rest of the people who have losing weight/building muscle as a new year resolution that's quickly forgotten after the months pass. You huv'ta stick to it.
Start off with cardio; brisk-walk/run (great for travelling 1/2mi), bike (better for 2mi+), skipping (great for indoors instead of paying loads for indoor treadmils, bikes, etc) and swimming (mostly do it on holiday)
Eat less crap sweets, sugar, excessive alcohol (straight spirits > red wine + sweet cider), fat too.
Buy some cheap dumbells. I might recommend getting them used as I bet loads of people will soon offload them 'cuz new year never resolution worked out for them.
Wouldn't bother with sit-ups, go for intense cruches. Pushups too. Do bicep curls.
Some good ways of getting bigger are squats, deadlifts, military presses, bench presses 'cuz you use a lot of weight and your very raped the next day but I must add that these can be unsafe unless you have spotters so it's a risk. Or use a small dumbell with plenty of weight.
Protein powder is good, usually cheap and good for after a workout probably a must if your a veg.
Some people do 3x (inc. abdominal) diff. muscle workouts 3x times a week and some people do 2x (inc. abdominal) diff. muscle workouts 5x times a week.
For chest, pushups, dumbell flys, dumbell presses -- all can be done on t'floor.
For triceps, french press, kickbacks, and the one where you lift a dumbell in the air from behind you.
For abs, crunches, some side ones too. Loads of others like reverse cruches.
For back, pullsups are probably the best here. Dumbell rows are good too.
For biceps, do curls, a lot of others too. (the biceps like being raped so I'd go heavy on 'em, start light still)
For legs, squats are good and lunges. Start with small weight as it's easy to overdo it and not be able to walk well the next day so be gentle to begin with also if you run/cycle time these two together.
I should mention the other three here. (squats, military presses, bench presses) but although these pwn, they can be dagerious without spotters, so maybe a gym-job.
Remeber breath too, if you grunt or whatever that's okay too 'cuz you are still breating. Go slow with proper form a bicep workout isn't a leg and back one.
BENCHMARK too, take pics, measure up and overtime (every xx month) check to see progress.