Weights-if you had to pick

With your time constraints you may want to think about skipping the weights and using your own body for resistance.  i.e, push ups, sit ups, pull ups, etc.  There are some amazing routines that def build strength with out the increased risk of injury that weights bring.  It's faster, convenient, and cheaper than the gym as well.  I am way too impatient to stand in line for a machine at the gym.

 If you work strength before the cardio you shouldn't have the atrophy you are talking about.  Through years of experience I have found doing the strength trng before the cardio is helpful...going in the reverse direction doesn't yield the same results for me. With that said...shoulders, lats, abs, lower back, triceps are all good for tri.

You might want to post this here:

http://www.beginnertriathlete.com/discussion/forums/forum-view.asp?fid=117#s

Good luck!

1) Run hill repeats.

2) Sprints and hill repeats on the bike.

3) 10 x 50 hard on :10 rest for the swim.

There, all your strength and core is taken care of in tri specific workouts!!

(Weight lifting does zero for improving endurance ...... just sayin' ......)

Oh .... and as Lauren mentioned, check the strength training forum for like-minded weight training responses.

strength training can complement endurance training (i.e if you dislocated a shoulder, you should keep your rotator cuff strong) but I am not aware of evidence that shows a correlation between weight training and endurance except fine tuning highly trained (like olympic) athletes. 

Workout A

Squats
Benchpress
Rows
Ab work

Workout B

Deadlift
Pullups
Overhead press
Ab work

All done free-weights.

A great total body program can also be achieved with Kettlebells.

I (infrequently) do a routine call "Extreme Cardio Kettlebell"  You will hurt!  But after the pain subsides you realize how much this could help you.  

Total cost involvement for a kettlebell and this DVD is prolly in the $120 range.  But do it once or twice a week for a whole body workout and you won't be disappointed!

Do a google search for  "Extreme Cardio Kettlebell" and you find a website called DragonDoor

I was also going to recommend kettlebells!  Kettlebells are a cardio and weight training workout in one and I've read that it can definitely help endurance sports such as running.

I think most of us triathletes are cardio junkies so we find all kinds of justifications to skip weight training.  It certainly can help you though.  Most people over look the secondary benefits like injury prevention, improved muscle efficiency and the extra calorie burning additional muscle provides.  For me I feel it allows me to train in my swim/bike/run at a higher level then my body would otherwise be able to handle.  So it may not directly improve my performance but by allowing me to train more/harder it does affect my performance.

To answer your original question you can get a lot done with just body weight training.  Lots of core exercises, lunges, squats, push ups, pull ups, etc...  I also have some resistance bands I use to work my hip flexors cause running and cycling do nothing for them yet weak muscles there lead to lots of injuries.

Nope.  It doesn't have any proven injury prevention or improved muscle efficiency for triathletes.  And I burn plenty of calories from cardio and the muscles I have without building up any that I don't need.  For me, it would hurt my performance because I would train the sports I want to get better at less.

This would cover you. Trade lat pull downs for pullups if you need to. Get instruction on doing the dead lift properly if you haven't done it.

Personally, when I'm in a hurry I use cable machines (Free Motion?) in a circuit. Chest press, row, overhead press, lat pull down, and squats. No rest during the circuit, rest, then do it again. Finish up with core, don't forget your lower back. As long as the gym is not crowded, it can be done in 25 minutes easy. Know your options, if the cable chest press is taken, grab some dumb bells and use a flat bench, try not to wait for anything.

It is physically impossible to "prove" injury prevention.  So it is easy to say that weight lifting does that!

Again, as Jorge mentioned, do it for your own personal satisfaction, to break up the monotony of other workouts, or to help rehab from injury or correct muscle imbalances.  Have fun with it.  But don't snowball yourself into thinking it improves your endurance or performance (again, unless you are doing it to rehab an injury).  To each their own!!

I do a 20 minute upper body circuit twice per week, with no breaks between sets.  I also do Pilates for core 2-3 days/week.

 Circuit A (x2): pull-ups, rows, French press (or tricept extensions)

 Circuit B (x2):  Curls, flyes, pushups

Circuit C (x2):  scapula rows, snatches, shrugs 

 *scapula rows help prevent rotator cuff/swimmer's shoulder problems

There are a lot of studies out there correlating strength training to gains in explosive events (sprinting basically), but not any conclusive evidence to increases in the endurance aspect.

Think of it this way.  What is going to be more beneficial?  Working your legs for 20 - 30 repetitions of differing weights over a period of a 1/2 hour, or working those same muscles 2700 repetitions over a half hour of cycling (90 rpms x 30 minutes) supporting body weight??

The benefits of increased weight loss are often shown with circuit training as long as you keep moving quickly and keep your HR up.  In per hour efforts, you will burn more calories than running or cycling.  If that is your goal, go for it!

And how can that be proven????

No you can't "disprove" it (though there are good reasons to disbelieve that there shoud be a connection). 

The only point I will make about your personal observation is that the best you can say is that it "appears" to help.  And I could just as validly argue that it is other factors that appear to help (like the fact that you are training more and you are gradually building up all the support your body needs to train harder). 

But if you "feel" it helps AND enjoy it, then by all means continue with it.  The only way it's likely to "hurt" you (assuming you train intelligently) is that you could likely be a better swimmer, biker or runner if you were actually spending the time swimming, biking or running.  For most, that's not a deal-breaker.

That's easy!

Lat pulldowns - Absolutely necessary for strong swimming
Calf raises - Your calves can be what's easiest to fail on the bike.  Strong calves are so often overlooked.
Squats (if they're done right) - Overall leg, power for the bike.
Leg extensions and hamstring curls (on opposite days) - again, power.
Back hyper extensions, side lifts, crunches and ab work - A strong and stable core is ESSENTIAL to ALL three, swimming, biking & running!
Dips - Shoulder strength and stamina really helps on the bike while riding aero.

Mix in some yoga and stretching exercises for fleixbiity.  Don't forget your ankles and neck.  You need range in your angles to get your feet flat with pointed toes for swimming and you need flexibility in your neck to survive cycling for several hours while aero.

Alternate flexor/extensor muscle groups on different days (leg extensions for thighs one day and hamstring curls for another) and don't lift the muscle group that your endurance training that day (no lat pulldowns on a heavy swim day, no squats on a long bike day).

I agree.  My point is that weight training might allow you to put in that extra half hour or extra interval session without getting injured.  In my own training I've seen enough correlation to know I won't be running above 50mpw without a consistent strength training program again.