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I look "soft", a little pudgy even, which also makes me look really young, like I'm 12 (I'm 18). But I don't have the money to get myself to a gym or a trainer anything and frankly, I'd be embarrassed to go. Can I get a "harder" look with just push ups and sit-ups etc?
You can, however it is exponentially more difficult to encourage hypertrophy with body weight exercises as doing any hypertrophy training under 70% of your one rep maximum weight is not only less optimal, but requires much more mental fortitude as psychologically people tend to reach a peak of stress on their nervous system, long before they have done enough work to actually encourage hypertrophy when maintaining physical tension under 70% of 1RM.
If all you want is to lose some weight, then yes, anything that gets the body moving burns calories. If you want to improve your body composition however and stimulate the growth of lean body mass, it is not only easier, but more efficient as well to use a gym, as you can easier track your progress and progress with linear progressive overload (adding 2.5lbs to each weight every 1-2 weeks), as apposed to trying to get in one extra pushup.