50 pounds in total, And No Push ups will not build muscle mass to a noticeable degree. The dumbbells and the weight will be much more affective.
You will definately need the dumbells, but to say to someone who is admitedly weak and who does not at this point "have muscle mass" that pushups will not help is not true. No, they do not work everything, but I do know people who developed amazing bodies with nothing but their body weight. Pushups, jumps, squats, planks etc. Push ups are amazing, especially if you do them right. Do them and enjoy it, when they stop working, worry about that then. ( you will note that everyone here has their own opinions, but mostly it is because of personal experience. Everyone has their own workout issues, and has worked with people with different issues. Ever BODY is different as I always tell the kids I have taught in the past, and the all the people I work with actually. Take everything you learn here with a grain of salt. Basically at this point you just want to make sure you are doing everything safely, properly, and learn what works for YOUR body.
To answer your original question: You want a home workout? Honestly, I would say do
1. A bit of core work, first (supermans, planks),
2. Jog in place to get yourself warmed up.
3. Do pushups until you feel your muscles burn, but not to fatigue (that's the first set, make sure you can do them properly, tight flat abs, do not let your back sag),
4. Do squats without weights to feel what it should feel like (chest out, butt back, not TOO much curve in the lower back). If you do this, tense up your muscles, and PRETEND you have weights. Tighten your whole core (like a corset around your spine), PRETEND it is hard to stand up, do it nice and slow etc. This is just to get used to the movement... then add weight slowly, I'll not presume to tell you how much weight you can lift right now, because I have worked with kids I thought could do 20 lbs on each side and they could only do 5-10lbs properly!
5. Back to the pushups, this time until failure (you feel you cannot do them properly)
6. Squat jumps (go down into a squat, still pretending you have weight, then as you come up quickly actually jump up, just a little at first and as you get used to it you can try to pull you knees closer to your chest.)
7. Dips, now do not go too crazy on these, start with your hands on a chair and your feet on the floor. Depending on the way you hold your arms in your push ups your triceps may already be quite tired. Tighten your abs and go down just a few inches to feel what it will be like to go back up. Next time you can go lower and lower, never until you feel strain in your elbow joint, but just muscles working. Do this to almost failure, then
8. More squats, or move on to lunges.
I do not want to throw too many new exercises at you because as you stated you are just beginning. The important thing is to work the whole body, increase the exercises as you get more comfortable, and do not hurt yourself. Oh, and do this every other day.
That said, is your family also vegetarian, or is this something you do on your own? I actually did a research study on it and wanted to prove it was not healthy, but proved myself wrong. As long as you get a variety (don't just not eat the meat part when other people eat a full variety!) then you should be fine. I don't know what you already know, so I cannot comment further on that. Before you go adding to you diet, take a note of what you are actually eating. I do not want to assume you are not getting enough protein, because many vegetarians do just fine in that regard. Protein aside, make sure you are getting enough actual calories. This is the part many vegetarians have the hard time with.
