I've decided to start off doing a basic strength program. I haven't lifted weights in a few years, so I have to work my way back into it. I currently have a set of adjustable dumbbells, with a 5 to 52.5 lb. range on each dumbbell (so 105 lbs. total max). I've been doing the following:

2 sets of 5 reps, 1 set to failure

Deadlifts
Bench press
Bent-over rows
Military press
Bicep curls (none of the others really hit these!)

I do the last set to failure because I can deadlift, bench press, and row more than the max weight of my dumbbells. I'm going to join a gym soon, so I can increase the weight.

For "cardio" I play "Wii Sports" boxing for 30 mins. or so. More fun than running!

In the future, when I'm in better shape, I plan on joining an MMA gym near me, AMA Fight Club. They have a few current UFC fighters training there (notably Dan and Jim Miller). I'm a huge MMA fan, and I used to wrestle in high school and college. It was a lot of fun, and I want to take it to the next level.

I have a chin/pull-up bar that I haven't used yet. My bodyweight is currently too much for me to pull myself up.

What works better lighter weights that you can do 12 to 18 reps or heavier weights where you can only do 3 reps or alternate on different days?

I think a variety of reps is best, 5x5 is fine for a novice/intermediate although IMO I found more volume to be better for squats.

The trouble with progresing on small muscles like biceps is that a 5# increment is usually too large.  You can either go to a wider rep range and try to progress that way or use microloading. 

I'm also going to add some Tabata routines on my "off" days (Tuesdays, Thursdays and Saturdays). I'll do sprinting and burpees (which I'm sure will kill me).

Here's an update of my progress. I've been using the barbell and weight bench at my parents' house instead of my dumbbells. This allows me to work out with heavier weights. Haven't tested the max weight I can lift, so these are what I'm doing 3 sets of 5 with:

Deadlift: 175 lbs. I could probably do more, but I'm steadily increasing the weight every workout anyway

Bench press: 165 lbs. I did this on Monday, and it was the first time I was actually challenged by the weight. Needed a spotter for the last rep or two. I repeated it on Wednesday, and had no trouble doing 3x5. Moving up in weight on Friday.

Row: 135 lbs. The weight bench I'm using has an attachment for rows, curls, etc. I've loaded it to its capacity at 135 lbs. I'll have to switch to the barbell for bent-over rows.

Shoulder shrugs: 175 lbs. I do these after I finish my deadlift set, keeping whatever weight I had loaded on the barbell.

Shoulder press: bodyweight. Still doing shoulder pushups. These are actually pretty tough. I go to failure on the last set.

Bicep curls: 110 lbs. Again, using the bench attachment for these. I think it's probably easier using this than if I did curls with dumbbells.

I finally added some sprinting to my off-days. I started on Tuesday, and managed to do half of a Tabata routine before I got winded. My brother is going to give me some competition to run against from now on, so I can really push myself.

Update:

Deadlift - 220 lbs. Also tested my 1RM, lifting 250 lbs. It wasn't tough, so my true 1RM is higher, maybe in the 275-300 lb. range.

Bench press - Haven't been benching. I need a spotter. I've just been doing as many pushups throughout the day as I can. Yesterday I did 126.

Rows - 160 lbs.

Front shoulder lift - 70 lbs.

Upright rows - 105 lbs.

Shoulder shrugs - 220 lbs. Still staying even with deadlifts.

Still doing shoulder pushups, and I added Australian pullups to my routine. I've also been sprinting on Sundays. I'll be joining an MMA gym sometime next week.