I wear undershirts daily. I tend to sweat when it gets warm, and they help keep me cool and from sweating through my outer shirt.
I would reconsider using the homemade soap. While many people see homemade soap as a tool to save money, in many cases you would save more in the long-run by using a good quality detergent. Modern detergents are far superior in removing dirt, stains and odors than homemade soap concoctions. Modern detergents also help your clothes last longer.
Better-quality detergents contain multiple enzymes to attack protein-based stains such as sweat, blood/body fluids/sweat, grass and many food stains. Liquid detergents help remove body oils that we leave behind on those garments that are worn close to the skin. They contain anti-redeposition agents that suspend soil in the water during the wash cycle and keep them from re-depositing on clothes. They contain bleaching agents that whiten your clothes and optical brightening agents that keep whites looking whiter and colors looking brighter. They also contain agents to keep your colors from fading as quickly. Tide and Persil are great detergents, but are often the most expensive unless you find them on sale ($0.25/load on average pre-sale price). There are other great detergents that generally work just as well for about 1/2 the price. Member's Mark liquid detergent by Sam's Club and Kirkland Signature detergent from Costco are rated by Consumer Reports to clean almost as well as Tide and they are quite reasonably priced.
The problem with homemade soap is that it contains real soap - something that doesn't do well in hard water and lower wash temperatures. In order for soap to be effective, you must have hot enough water to allow the soap to get into the fabric and release dirt and stains from clothing. This takes a lot of heat and a lot of water to do it effectively. Soap works better in top-loading machines than it does in traditional non-HE front-loading machines because they use so much water. You must also rinse your clothes aggressively in order to get all of the soap out of the fabric. This requires 2 - 3 hot (or at least warm rinses). Most washing machines use cold rinses. If you use a cold rinse directly after a warm wash using pure soap, the fibers of your clothing constrict and trap in the soap and all of the gunk that was removed from the fabric during the wash cycle. This can cause clothes to become yellow or gray over time, and have a musty or "BO" smell.
Water temperature is also a major factor in cleaning clothes. Cold water washing is quite popular these days due to cost and/or environmental concerns, but it cannot always get the job done on tough stains and smells. Normal human body temperature is 98.6F. It's at this temperature that body oils and sweat get stuck into our clothes. If you are washing these garments in cold water that ranges from 45 - 65 degrees, it is going to be difficult to remove these types of stains from clothing. The warmer the temperature of the water, the easier it is to remove stains and general nastiness from clothes.
I really like the Kirkland undershirts from Costco. They are normally priced at $19.99 for a six-pack, but go on sale twice per year for around $15.00. The ones I'm wearing now I've had for almost two years, and they are still in great shape!