How can I find places to store/land a private airplane?
The first thing you usually do as a pilot is to look at a sectional chart, which shows all the airports and airspace in the area you want to fly (and transit). They look like this:



You can see that the Orlando area offers a number of airports. I would avoid anything with the word "International" or "INTL" in the name. Mostly these airports cater to commercial traffic; GA traffic is allowed but usually requires a lot more work to get you in, especially if you are only VFR (not instrument rated or on an instrument flight plan). 

You can see Orlando Apopka (X04) in the upper left corner, but also available are Executive (ORL) just above Orlando International or Kissimmee Gateway (ISM) in the lower center/left. If you want to make sure the airport has fuel, these are denoted by the "ticks" around the airport symbol:



If you want to look at more sectional charts, they are offered for free on sites such as Skyvector which also has flight planning tools. 

To answer your other questions:


  Which brings me to my question, if the pilot of a small aircraft wants to land in place far yonder what's the best way to plan for this?


Flight planning is part of your initial pilot training. By the time you earn your PPL you will have planned a number of cross-country flights (albeit not that far as Texas->Florida). It starts out by picking an airport and looking up the information in the chart supplement along with one or two alternates. You then call the Fixed Base Operator (FBO) and ask about fuel rates and overnight parking, and then continue on with your flight planning tasks (like checking weather, routing, etc). 


  Is it common practice to call all the "small guys" in the area first and see if they're willing to work with your flight schedule.


Calling the FBO you intend to visit is advisable, especially if you need to use services like parking or crew cars. This lets them know to expect you and they can tell you what they charge. Some smaller airports won't charge for ramp tie-downs if you buy a certain amount of fuel, etc. Some airports don't have an FBO so they are "self-serve", meaning you pull up to a gas pump just like in your car and fuel yourself.


  If they gave you the go-ahead to land and store your aircraft in their facilitates is there any paperwork you should sign to protect your aircraft in the case of theft/vandalism?


Leaving your airplane at the airport isn't like parking it in the parking structure. Airport access is tightly controlled, especially at airports that have FBO's or operating control towers. Some smaller airports don't have that and if you know how to find the codes you can pretty easily get through. Either way though, vandalism/theft is usually a non-issue. The biggest worry you have is weather. If you are really concerned about your aircraft, you can ask the FBO if they have a hangar and if there is space available. Storing your aircraft in the hangar can be very expensive. Typically you just tie down on the ramp and put in some sun-shades.


  What if the commercial airport is the only viable "runway" you see close to your target destination? What red tape is involved with landing/storing a small aircraft on commercial landing strips?


I assume you mean a major commercial airport like Orlando International. The only "viable" option to me denotes an emergency, as you should have planned to fly somewhere else and only in an emergency would you want to land at a major commercial airport. If you need to declare an emergency and land, ATC will clear the airspace and make you the only user of that airport. As for storage when you get there, many airports even the size of Orlando International have a GA area where you can tie-down, but getting fuel and even the ramp space will be very expensive.


  Say I'm the proud new owner of a Cessna 172.
  
  To celebrate I want to fly my family from Texas to Orland Florida for a weekend visit.


Just FYI, a Cessna 172 is not a "family trip" aircraft. Two adults, one kid, and baggage will pretty much max out the aircraft. You can remove fuel to add more weight, but then you are puddle-jumping. 


  Local takeoff is easy enough. I live close by, and I know the locals so I can "borrow" a runway.


Unless the airport is marked private (PVT) you are free to use it and don't have to "borrow" anything. Only airports marked as such in the chart supplement will require prior authorization to land there, so the majority of airports are "public use" and you can just jump in your plane and fly there without warning (as long as the weather is above minimums).