I recently started weightlifting again after taking 6 months off. Now,
when I do bench presses or tricep excercises, my elbow joints
frequently become sore. This didn't used to happen. If I push myself,
the soreness can last for days afterwards. Sometimes the joints pop
while I'm doing the lifts. Similar happens to my knees when I do leg
presses. In both cases, I've always kept the weight light enough that
I can do at least 8 reps. Should I be worried about the soreness and
popping?

You shouldn't ignore the ache.  One time I tried to experiment with
leg presses by doing reps with my toes pointed in and out as well as
the usual pointed straight ahead.  I started getting aches in my
knees.  I stopped doing the reps with my knees anything but straight
ahead, and the ache went away.  I suggest that the wieght that you are
lifting is too much for your joints when at an angle.

PbA

peggy_bill has a good point. Maybe that's my problem with my elbows
too. Does anyone know the correct direction to point your elbows when
bench pressing? Should they be sticking out perpendicular to my spine,
or angled downward?

It's normal to feel aches and pains in your joints after working out
for the first time, or after a long period of inactivity.
   More than likely the pain you're experiencing is your tendons
stretching in ways they aren't used to. They will eventually adapt to
the new stretching and weight demands placed on them, and the pain
will go away. Don't press it too hard during this period or you may
permanently injure yourself. Eight reps should be fine, but be sure to
use proper form.
   Remember: never EVER lock your joints when exercising. You will
hyperextend them, essentially forcing the joint to go in the opposite
direction it is intended. Over time this will cause permanent damage.

   Also, when doing the bench press you want your elbows parallel to
the bar. Keep your back flat on the bench. If you have to bend your
back to lift the weight, then try a lower weight.

One last thing:
There's a big difference between pain and soreness. Soreness should be
tolerable and not prevent you from normal activity. Pain will be sharp
and may prevent you from activity. Pain is a signal to give your body
a break, soreness is your body adapting to doing something it's not
used too.
 
Listen to your body and use your best judgement!
Good luck!
-WILL