I am your size, so I understand the concerns.
A good set of semi-slick tires will be good. I would go with almost any size tire that the rim will accommodate. The tube is pretty irrelevant when it comes to weight bearing, so just a standard tube with the appropriate stem for the rims, and the proper size will be fine.
I have spent more time trading tires trying to find what I like most than anything to do with weight bearing. I prefer a 1.5" or so tire, with a little tread for when I am riding in gravel etc. I also look for a tread design (rib etc) that allows quick rolling on pavement.
I would second the vote for spending a little bit on tires. There are some good inexpensive tires, but there are some great mid-range tires. In general, high tpi casings provide a nice supple ride. I currently ride 700c wheels on my hybrid, so I don't have any guidance for 26" tires, but I am sure there are some good ones out there.
I had someone who rides a hybrid with stock tires ride my bike and they said that they felt like they were flying. The only significant difference were shifters, rear derailleur and tires. The shifters and RD provide no significant benefit, so guess what accounted for the difference that they were feeling?
So, tire choice can make a difference. But anything other than super light weight racing tires will hold your weight if that is your only concern, Just keep the pressure up to near the max. I usually do max pressure for the rear, and 10% less for the front.