If you want to gain muscle, you need to do cardio like running/swimming because that's what rips the muscle and sit ups and other strength building exercises build it up. You need to do both to see results!
You're off to a good start. A month is no time at all. Don't give up. You need a lot more time and effort put into learning about your body, and nutrition, and trying things and tweaking them to get the results you want.
Getting visible abs is mostly a matter of getting your body fat percentage down below what your body naturally tries to maintain.
This is usually most effectively accomplished through a structured, disciplined diet, combined with muscle building exercises and lots of fat burning cardio and HIIT.
If you just eat 'mostly healthy' but don't have your diet and exercise basically scientifically planned for the necessary fat loss, it's next to impossible to gain visible abs.
Start reading up on fat loss through diet. Look into balancing your macros.
There is a lot to learn and you just gotta start googling.
When trying to shred fat and gain muscle definition, you really have to sweat. Within a month straight of working out, im getting closer and closer to having my dream stomach!
For my cardio i love running, cycling, swimming or dancing or zumba workouts. But usually i will do HIIT. There are HIIT exercises on here, or if you search HIIT workout videos you can find some really good ones on YouTube.
I try to workout 4-5 times a week, and one one day i will do a "cardio burn out" and will do HIIT along with a run or a bike ride! :)
So, lots of cardio, and do ab exercises so you can shape your ab muscles! And again, always try to eat clean and get lots of fruits and veggies into you! :)
Check out the Nike+ app there you will find loads of strength excercises and also look at my liked workouts maybe they can help you.