Whether you should do cardio before or after strength training is an age old question, and one without a concrete answer. Really, you should prioritize the workout that is most important to you.
If you're training for a running event, for example, you'd be best off doing your cardio when you're fresh and leaving the push-ups and sit-ups for the end. When weight loss and strength gains are a goal, the calisthenics should come first, along with other weight work.
The Importance of Order
The order of your exercises impacts your results. When you first hit the gym, you're at your best. Your energy stores are full and your attention is primed. When building upper body and core strength are goal No. 1, hit the push-ups and sit-ups first. You'll be more likely to fit in all your reps with quality if you do them before you're tired from cardio.
Do realize, though, that to truly define these areas, it'll take a comprehensive strength-training routine done two to three times per week — the sit-ups and push-ups are a good start, but won't give you huge results in the long term.
Form is likely to be better for the workout you do first, too. If you fatigue yourself with a long cardio session, you may struggle to perform push-ups and sit-ups accurately. When form falls apart, so does the exercise's effectiveness.
For the guys, order can affect your hormones. A small study published in a 2012 issue of the Journal of Strength and Conditioning Research found that it might actually be better to do the cardio first. Doing aerobic activity, or cardio, before strength optimizes testosterone response. Note that the cardio done in this study lasted just 30 minutes and the strength training included four exercises using heavy weights, not bodyweight exercises.
The Convenience Factor
Before you do push-ups and sit-ups, it's a good idea to be warmed up so that your muscles get the fullest range of motion and you're less susceptible to injury. Doing cardio first serves as double duty — you get your aerobic workout in and you warm your body for the floor exercises.
When you do push-ups and sit-ups first, you'll need to precede them with a short 5 to 10 minutes of light activity, such as marching in place, to be warm enough for you to really put forth optimal effort.
What's Right for You?
If you're in relatively good shape, even multiple sets of push-ups and sit-ups are unlikely to be taxing enough to cause fatigue in your lower body, the area that's most often used for cardio. Thus, you might not really need to worry as to whether these exercises will drain you to the point that your cardio suffers if you were to do them first.
Now, if you've got a habit of going to the gym with grand intentions of doing both cardio and calisthenics, but lose steam halfway through and ditch whatever is left on your plan, start with the push-ups and sit-ups so they don't get left off when you're tired at the end.
Another bonus of hitting the push-ups and sit-ups first? They'll likely feel easier. Strength moves after cardio are usually perceived as more challenging.