"what I can use in terms of wheel sets"
The mismatched rims suggests that one rim was damaged and replaced by a prior owner. There is no particular need to replace either rim, a mismatch doesn't hurt anything. If the rims are true (straight - spin the wheel while watching the gap between rim and brake pads) and no spokes are slack or broken, you don't need to do anything. Still, mismatched wheels can bug some people (like me). If you want to get matching wheels, you should have a good wheelbuilder at a local bike shop rebuild the wheels with the existing hubs (assuming the hubs are matching), new quality rims, and new spokes/nipples. Cost perhaps $260. A properly handbuilt wheelset will be better than a new off-the-shelf wheelset of equivalent price. I like Mavic Open Pro rims (high quality, strong, not too deep section, looks right for the bike) but there are other quality rims. Spend the small extra for double-butted spokes.
"casual clipless pedals/shoes"
There are shoes for clipless pedals that have a sort of retro/classic look, with laces and leather, if that is what you mean. Like the Giro Republic. There are also shoes for clipless pedals that look like sneakers or even flip-flops, certainly "casual", but it is debatable how good they look.
For pedals, SPD-type pedals are simple and the cleats are recessed into the shoes' soles so you can walk normally. The lightest SPD-type pedal may be the Wellgo MG-8. There are also heavier SPD pedals with the binding on one side and a standard cage on the other (so you can ride in regular shoes too), like the Wellgo WPD 95B. Those are both about $40-50 the pair.
"Would a newer seatpost help shed some weight?"
Minimal. Maybe 50 grams (2 oz) at best if yours is already aluminum.
"I do ride with friends and it seems like the guys on newer bikes crush me on the uphills. My bike weighs about 21lbs or so."
Bike weight makes some difference, but the main factors are technique, legs/lungs, gearing, and rider weight. 21 lb is quite light. The difference to your friends' bikes is just a full water bottle or two.