(I'd like to "bulk up" a bit in terms of muscle) but at the same time I'm probably still 15 - 20 lbs overweight (I weigh 215 and am 6'5" -- definitely not enough of that is muscle ;)).
You are in no way, shape or form, overweight at 215 and 6'5". You may have some "pudge" but dude, you are in good shape compared to most people I see everyday. (Now if you were 5'10" and not muscular, I would think differently.)
Ideally I'd be doing free-weights and following a program similar to the one I've read about from Mark Rippetoe, but that will have to wait. The gym I'm at just isn't set up for it. Lots of machines, some dumbbells and a Smith Machine is about it, plus a huge array of cardio machines -- I prefer spinning (even without the class).
You can do body weight stuff, and teh smith machine is not the devil. If it is your only option, better to do some lifting than none. Just move to a gym with free wieghts ASAP.
My quesiton: Can I "bulk" -- so to speak -- I'd really like some muscle for the first time in my life -- I've never done a pull up, for example -- and at the same time lose the extra body fat?
As a rank beginner, yes. After some time under the bar, not so much.
At your age I would be cautious of your diet first and foremost. Get what your eating on a daily basis down pat, and you may not even have to worry about cardio at all. At 44, you probably don't want to mess around with gaining fat in order to gain some muscle. At 24 I would say, hell yeah "squats and milk", but at 44 it's more like "fish, spinach & squats."
I've been trying to follow the science, here, but it's confusing. If my goal is to get my BMI well into the healthy range (it's hanging out at 25.5, now) and build as much muscle as is simultaneously possible, what does my ideal workout look like?
1) Pay more attention to the mirror than your BMI. Muscle weighs more than fat. I believe I am in obese range, and trust me, I look better than most 240lbs people walking around.
2) A routine that focuses on the big compound movements with progressive overload is the way to go.
Oh, and what's the quickest way to go from not being able to do more than a couple really good push ups to being able to do *pull ups* at my age and how long is it going to take?
The quickest way is to do more pullups
. Lat pulldowns and assisted chins will help too.
How long is it going to take? Depends on a lot of factors, but the harder you try, the better you eat, and the better you recover means the faster you get stronger.
Welcome to the "over-analysis-because-I-lift-weights-club" the view rocks from here.
But yeah, get your diet in check before anything if you can't get to a gym with free weights. And if you like it, ride that bike.