If you're looking to do 3-4 sets per week of one exercise then I think you'd be better off just playing more intensely with your dogs. That's way too low of a volume to even matter. Maybe get a weighted vest when you take your dogs on a jog and you can kill two birds with one stone?
I would suggest aiming for 10 sets per muscle group per week. The types of exercises I would focus on are called compound exercises (think deadlifts, squats, cleans, pushups, pullups). They work out multiple muscle groups and give you the most bang for your buck. Deadlifts are probably one of the best 'core' workouts and would improve your posture and back strength. As long as you're properly warming up, using good form, and not ego lifting they're safe. Compound exercises can be done with barbells, dumbells, or bodyweight and that depends on your goals. For wellness, I think your body weight is fine to start with. I would add in weight for bigger lifts (deadlift, cleans, squats) once you're accustomed.
The reason I mentioned CrossFit is that the workouts are typically timed, use many compound movements, encourage "functional training", and they're hard; the whole community has that push yourself one more inch vibe. For example, the CrossFit workout "Cindy" is:
AMRAP in 20 minutes
- 5 Pull-ups
- 10 Push-ups
- 15 Air squats
No matter what, this workout will take you 20 minutes. That's great if you're pressed for time; which it sounds like you are. You can then score yourself based on the sets completed: 11-12 (Beginner), 13-17 (Int.), 19-22 (Adv.), 24+ (Elite). This workout requires no gym, no weights, and you can do it at home with your dogs (or even in a park).
If that's too easy, the workouts can scale up ridiculously. Here's a good one called "The Chief" which, again, would only take 20 minutes:
AMRAP in 3 minutes
- 3 Power Cleans (135/95 lb)
- 6 Push-Ups
- 9 Air Squats
Then Rest 1 minute
Repeat 5 times