I can completely understand your logic. I grew up around power boats, so the whole idea of a 9.9HP, or (as was on my Catalina originally) an 8HP pushing a 25' boat seemed crazy. But the 8HP Honda and 9.9HP Tohatsu that I had on that boat pushed her to and above theoretical hull speed at about half throttle. That gave me a lot of reserved "oomph" for if I was fighting a headwind or a current. For reference, my nearly 11,000 lb Allmand has a 16HP diesel inboard.
So, having sorted some of the confusion out, Jones's questions are perfect, as are some of your answers. A $15,000 budget is a reasonable budget. On a smaller boat, like a 22, that will get you something new-ish (maybe into the 1990's), on a 25-27, that will get you something with a 1980's vintage.
Unfortunately, what you're going to struggle with (and we all do) is finding the boat that truly meets your needs. One of the tough things right now is that, as a novice (again, I was there 2 years ago, too) you don't REALLY know what you want. You may think you do, but with all due respect, that will likely change.
Trailerable boats are great, and there are a lot of people for whom they are a perfect fit. The idea of being able to head off to different sailing destinations on a moment's notice is cool! As is the ability to travel REALLY far (e.g., across the country, or even down to the US) with the boat is pretty neat. A trip that would take many days in my boat is a relatively short drive away (for example, it took us 3 days to go about 200 miles when we did the delivery of the Allmand a few months ago). That significantly opens up your sailing area. Plus, with the boat on the trailer, you have her close to home to make repairs/maintenance, and you don't have to pay storage fees or slip fees (assuming you have the room at home).
But there's also a down-side to a trailerable boat. The first is the time you'll need to set her up and to break her down. Someone mentioned 45 minutes for a 22' boat, and that's generally what I've seen others say. For a 25, I typically heard an hour, once you're good at it. So, your typical day will be get up, hitch up the trailer, inspect the tires, brakes, etc., then tow the boat to the lake/ocean. Find a ramp that is deep enough for you to launch (depending on keel size/type, this may not be as big of an issue), then get in line to launch. Spend an hour rigging the boat. Stow everything. Launch the boat, then park the tow vehicle. Go sail, make sure you're back in time to have daylight while you get your car and wait in line to pull her from the water. Haul the boat, wash her down, break everything down (again, about an hour or so). If there is a bad turn in the weather, you're stuck breaking it down in the rain/wind. If something happens at the ramp, you're stuck in line for a long time. Then you get to drive home while towing a long, sometimes awkward load behind you. Then you have to park the boat in her "home".
If you're day sailing, you're looking at losing easily 2-3 hours of your day doing the "trailer thing" and that may get old quickly. That's why many trailer sailors go for the weekend, or longer. But (generally) the smaller the boat, the less comfortable it will be to sleep aboard. The bigger the boat, the harder it is to trailer and launch. As you'll hear here a lot, in boating, everything is a compromise.
Some trailer sailors LOVE it, and never want to give it up. For some, the idea of setting up and breaking down the boat gets old, and they look at other options. One is, of course, renting a slip. That's typically the most expensive option, but if you want shore power, it's really the only option (there are lots of threads on solar and wind power, if you're interested). The next less expensive options are moorings and keeping the boat rigged on a trailer at a marina/boat yard. Given what you've said your goals are in a boat, if you decide to stick with a trailerable boat, this last option may be advantageous. You find a marina in a place where you'll sail most regularly and you keep the boat out of the water when you aren't using it, but still avoid having to break it down and set it up every time. It can also be broken down and towed to other venues when you want to. You still won't have shore power, and if you overnight you'll need to anchor or rent a slip, but it's a nice compromise that lets you get on the water faster.
What I'd respectfully suggest, though, is that you don't know enough to start to understand the compromises involved, and your best bet is to go look at boats. And I don't just mean one or two, I mean 15-20. Look at 18'ers, 20's, 22's, 25's, and if you can find them, a few 27's. Look at what you get with the different sizes, and decide which size best meets your anticipated use. Sit in the 22's cabin, and imagine the two of you (or more, if you have kids) sleeping there and then waking the next morning; how do you prepare breakfast? How do you handle your morning routine (shower, shave, teeth, etc.)? Where is the head located; does it offer enough privacy for your wife? What if one of you needs to "go" during the night, does it require waking the other person before the head can be used? Sit in the cockpit - is it big enough for your wife to sit comfortably while you're working the tiller (since most boats in that size range don't have wheels)? If you plan on having guests regularly, is the cockpit big enough for everyone?
Once you've been aboard a bunch of boats, then think about your intended use again. As I said, for some a 22 is a PERFECT boat. For me, with my family, there's no way we could make that work. Although we liked the IDEA of trailering our boat, the time lost rigging and breaking down the boat was a non-starter for us because we have 2 young kids. They would be going nuts during that time, and it wouldn't wind up being an enjoyable time. Eventually, we accepted this fact and went with a slip, and then went with a bigger, more comfortable boat. A trailerable may be just the ticket for you, but I want to make sure you have a good sense of what will be involved, because the last thing I want is for you to buy something that becomes an albatross.
All that being said, go back and read Denise's first post - I think she covered almost all of that, and in far fewer words than I did.