Getting Started with fitness

Hi! My husband and I are both looking to start a 2-3x a week fitness routine. We have a pool with a crappy gym in our neighborhood that we would like to join. We had the idea that we could go together with our daughter and take turns going to the gym while the other plays in the pool with our daughter. Ideally, we would each do a 20 minute workout in the gym and get a good swim on. We are both of normal weight (I am looking to lose a little weight while my husband is looking to build muscle.) We aren't in the best shape and my husband has a slipped disc in his back that acts up from time to time (part of why he wants to work on his core..) When I look up 'getting started working out' I am pretty overwhelmed. Any one willing to share their routine or point me in the right direction to get started? I don't even know hat sort of things to do - the gym has a ton of free weights, so we were both thinking some sort of weight routine for arms/core, the some crunches/push ups? While we have both worked out in the past, we feel like total newbies getting going again!