I've been able to get my schedule so that I go to the gym three times a week for the past two months. I've gotten some experience working out mostly with dumbells. I feel like I've gotten a but stronger, but I don't know if I'm really making progress fast as I should be. How should I be programming and planning my workouts for long term improvement? What resources are there for learning new exercises and building a plan that evenly works various muscle groups? I have access to a personal trainer at a reasonable price and I want to take advantage of that, but I don't think it's financially sustainable long term.
If strength is what you want, the most important things are weight (what can you do 3 sets of 6 with?) and getting good sleep.
Anything that gives muscles two days of rest and covers all the muscles you want to do will work. The three days a week plan you're doing is fine for that. If your workout is faltering because you're running out of energy (ie: you do better earlier in your workout even if you reorder your stuff) you may want to do more days and splitting muscle groups (commonly arms/legs or front/back).
The way I learned it is push/pull, one leg/two leg, bend/twist. It's fairly thorough; I've only ever been surprised by a weak muscle group once (one covered by side lunges). If you'd like to extend that, I'd make push/pull separate for chest and arms.
There are various other things not covered by this (forearms, calves, neck muscles) but they're largely vanity muscles. If you want vanity muscles that's cool too, but you'll need to ask about the specific ones you want.
A personal trainer is a great resource for learning new exercises. You can ask for recommendations for a specific muscle group, or pick out exercises ahead of time and ask them how to do them well.