I'm far from an expert, having just gotten into weight training myself this year...

I think this plan is much better than not doing anything, but unless the pushups are done with weight or position so that you can only do no more than 6, you're working more on muscle mass building or endurance rather than strength.

Once you've done about 15 reps in one set, then you're building endurance and tone more than strength.

From the Weight Training for Dummies book, I've learned these general guidelines:

Endurance
12+ reps
30 sec max rest between sets
2-3 sets
increase reps progressively

Size
6-15 reps
30-90 sec rest
3-6 sets
lift weight to near failure at last rep
increase weight 10% once you can lift 15 reps to muscle failure

Strength
6 or less reps
2-5 min rest
2-6 sets
lift weight to near failure at last rep
increase weight 10% once you can lift 5-6 reps to muscle failure

Edit:
Pushups aren't going to help build the muscles for the activities you mention as well as more specific exercises done with free weights or a machine. I'd get a set of dumbell bars and plate weights so that you can adjust the weight as you progress. A machine would be convenient if you have the money and space for it.