A typical lat day for me consists of:
Pullups 12 x 3
Palms out, chinups are palms in and work more biceps. No additional weight, use as a warmup.
Cable rows 12-15 x 3
with a gunner's grip, use your body, lower back for the first part of the motion and that last bit is where the lats really kick in. Your biceps should never get sore, if they are tone down the weight and concentrate on flexing the lats as you are pulling
Bentover barbell rows 10-12 x 3
I use a wide grip here and have my palms facing down, use palms up if you are doing biceps the same day as back, which I don't suggest. Again if your arms are getting sore before your back, lower the weight and focus on flexing the lats, there is no reason to extend your arms fully, in fact it will likely cause you problems with your tendons on the inner elbow. Try to keep the arms around a 130°-150° starting position, pull back hard and tight, flexing the lats on each rep.
T-Bar Rows 12-15 x 3
I personally don't use T-Bar because it can be done more effectively with an olympic bar and a gunner's grip underneath and I load the weight up on one end with the other end secured underneath something. You have probably seen people do it before but if not it's kind of hard to explain. Do these like the bentover rows but with more weight and more force, Your legs and lowerback play a big role in this lift as well, an excellent compound workout.
Dumbell rows 12-15 x 3
These will finish you off well. Allow the weight to be fully extended so you stretch your lats on each rep. Bring the weight up with a slight outer row, meaning not straight up and down, but like you are throwing a hook in reverse. Again, flex the lats as you bring up the weight.
I hope this helps, it has worked well for me, there are certainly other exercises you can throw in as well but these are my core lifts.
