Newbie Training/gearing Question

citrone

Am new to cycling but not to working out - do alot of pretty hard core eliptical work. Riding with my better half is not very challenging - although as new empty nesters, the outings (12-15 miles) are very enjoyable. We bought pretty inexpensive bikes to start - just to make sure we would enjoy ourselves - I believe we bought Trek 7300's. They were on sale, had 24 gears, and since neither of us had been near a bike for 20 years, they seemed like reasonable first steps. With all of my eliptical training, I occasionally love to ride standing up ( my wife thinks I look like a dork while doing this ). I can go faster and dont have to worry about "bike butt". However, whenever I do this, my speed goes way up, and my wife is left in the dust (that may or may no be interpreted as a problem). So, I now find myself making solo trips, with speed and a good workout as my objective. My question is this, when I'm in the top gear which I call 3/8, and get up to over 20 mph, I find myself wishing I had additional gears.
I dont know anything about cycling equipment, chains, deraillers(?), gears, etc.... Is there any kind of adjustment I can make to my equipment to alter the gears? Do I need a different bicycle? Can my existing bicycle be altered to change the gears?