Yes, it will save water. How many people in the house and how much of the day are there there verses say at school, the office, or elsewhere? How much does the current toilet use? The really old ones could use as much as 8 gallons/flush. The newest dual-mode flush ones use like 0.8gallons/1.2gallons, depending on the mode. So, depending on how many people and how long they are home, many hundreds to thousands of gallons per year. In many places, that affects not only the cost of water, but also the sewage charges as well.
Look at the Toto Drake II as a starting point. Some of the choices are style, height, single-piece, two-piece, skirted, 'traditional', elongated, round. Toto is the largest toilet manufactuer in the world, and has a huge variety of toilets from several hundred dollars, to many thousands, depending on style and options.