Welcome to the forum and thank you for the great questions. These are some of the exact question rocket scientists and engineers working for organizations like NASA ask and attempt to answer all the time.
What kind of materials are there to use that are lightweight, sturdy, and relatively inexpensive for the kind of deal you're getting?
Currently aerospace industry uses mostly aerospace grade aluminum alloys. This really just means that it is really high quality. It is light weight and strong and good for the structure that makes up the body of the spacecraft, known as the Bus. Exotic materials, such as carbon fiber may be used as well but tends to be much harder to form into structural parts. But where weight is a factor or electrical connectivity is a concern, one may pick a more exotic material.
What kinds of fuels can you carry in bulk supply that are efficient enough to propel the craft or object into space without much consumption?
There are three terms you should familiarize youself with when talking about engines and their efficiency. First: Thrust is the amount of force an engine provides. It is caused by the expanding gas of the burning fuel exiting the nozzle. Second: Total Impulse. It is the amount of push an engine will give during a firing. It is equal to thrust times time (i.e. an engine that provides a thrust of 6 newtons and burns for 2 seconds has a total impulse of 12 newton-seconds. Third: specific impulse. This is a measure of how efficient an engine and fuel are. It provides a measure of the change in momentum per unit mass of fuel. It is equal to the total impulse divided by the mass of the fuel times gravity. The higher the number the less fuel you need to achieve a certain velocity. However, not always with the same thrust. For example, the space shuttle main engines had a specific impulse of about 300 seconds. They provided lots of thrust for a short amount of time and used a lot of fuel. A plasma thruster (like a hall-effect thruster) may have a specific impulse of 3000 seconds but have very small thrust, but burn for a long time and use small amounts of fuel
Would it be more cost-efficient to make it small and to support just a single person?
The thing to remember is that form follows function. The spacecraft is designed to perform a mission and the size and shape supports the function. The space shuttle had a space transportation mission. Actually the humans were a small percentage of the overall weight at liftoff.
Is it possible to make it a vehicle capable of quick travel into space and back?
See notes on thrust and impulse. To achieve orbit you must reach a certain velocity, then your movement is governed by orbital mechanics. For high acceleration, you need lots of thrust quickly, this requires lots of fuel, which requires even more thrust to accelerate (F=ma). It becomes a losing battle.
Is something along these lines already being researched or designed?
Check out work on exotic forms of propulsion such as ion thrusters, laser propulsion and solar sails.
I hope this helps.
"As the circle of light increases, so does the circumference of darkness around it."
~ Albert Einstein