Alexander,
IMO you don't need to join a gym. I'm not a member of one and manage just fine. As you mentioned, you can likely get a complete workout in the time it would take you to get to the gym and back.
Here's what I recommend:
First, make a list of your resources. Do you have hiking trails, a playground, park, lake, stadium steps, track, mountain/hill, etc, around?
Do you have a bike, access to a pool, jump rope, weighted objects of any kind?
You can get a great workout almost anywhere with a little creativity. If you have a playground near by walk over – carry the baby on your back to increase the workload, once there do a variety of bodyweight exercises – pullups, pushups, air squats, jump squats, lunges, planks. You can get in incredible shape using bodyweight exercises alone.
For cardio, hike, bike, run. Depending on your goals, cardio does not have to take hours. Try doing multiple sets of air squats followed by a 200 meter sprints (thank you Dan John) for 10 or 15 minutes – set of squats/sprint/squats/sprint/etc. It will kill you and takes almost no time at all and no equipment either.
With all the books, gyms, and fancy equipment out there it's easy to lose sight of the basics and make training more complicated than it needs to be. Again, borrowing from Dan John, all you have to do is pick up heavy things and carry them and press them over your head. Cardo is even easier – walk, run, climb stairs, hike, ride, swim, sprint. It's all good.
If you have money and want to buy some equipment I would recommend a pullup bar first, jump rope second, cheap weight set third. As Coach Ilg has said many times, the pullup is an incredible exercise that will build mental and physical fitness like few others. With a jump rope you can get a great cardo workout in very little time. The weight set I use I got on Craigs List with a bench and rack for $150.
Finally, you're here so I'll assume you are familiar with the Ilg Master's body of work. I would recommend that you also check out Steve Maxwell's site as he is all about bodyweight exercises and getting workouts with little to no equipment wherever he finds himself. I would also recommend you check out Dan John's site. I'm reading his book "Never Let Go" right now and have followed his writings for some time. The man is a Godsend of information and talks a lot about getting a workout without needing a bunch of fancy equipment or gym memberships.
Good luck!