I support both of the other answers given so far, but given your clarifications, I'll add an answer of my own.
To answer the question "Is your current strength building workout good to start off with", the answer is no.
You say that your goal is strength training and to not get huge muscles. Luckily, you can do this. The rep range that you need to work out in to get strong is 3-5 reps at relatively high weight. The rep range that you need to work out in to get big muscles is in the 10-15 rep range. Your current workout plan will give you big muscles and not much strength, especially since you're only working out your biceps, a relatively unimportant muscle.
I've done a program for three months (much like the one Berin suggests) focusing on squats, deadlifts, bench press, and overhead press. In those three months, I've gained 15-20lbs of weight, increased my strength in these lifts 25-80%, depending on the lift. And, my muscles are not noticeably larger at all. People wouldn't notice that I've been lifting. I am a lot stronger, though. You can do the same thing.
Even if all you have is dumbbells, at least focus on the lifts that matter for strength. Overhead press. Bench press. Squats (look at the goblet version). A deadlift-like thing. I'd also suggest investing in a chin-up bar. They're super cheap.
- Overhead press will strengthen your entire shoulder musculature and triceps.
- Bench press will strengthen your chest, shoulders, and triceps.
- Squats will strengthen your lower back, abs, glutes, hamstrings, quads, and calves.
- Deadlifts will do the same as squats, but you can do them with heavier weight, and they focus more on the back.
- Chin-ups will strengthen your upper back and biceps.
That's what functional strength training is about. It's not about size. It's not about biceps (although they'll get stronger). It's about multi-joint, multi-muscle movements that allow you to move a lot of weight, and move more weight the next time. Doing a full body routine like this will also reduce your body fat, eventually showing off your abs.
You've mentioned arm wrestling a few times... I guarantee you if you just do your current workout, you will not be winning arm wrestles. You will need chest strength. You will need shoulder strength. You will need upper back strength. To avoid injury, you will need strong supporting musculature in the shoulders, and strong ligaments and tendons. You will need a strong lower back and abs so that your core can be a stable base for you.
This will be true for almost anything that you need strength for. It will not come down to a single muscle.