Hello Trips,
You cannot compete with the costs of the airlines for air travel, even if you do own your own plane. You fly ganeral aviation (GA) because you love it, and it takes you exactly where you want to go, when you want to go (and you can carry your pocket knife onboard!). Not to save money over commercial air travel. That said, for distances in the 100 mile range, which are not served by commercial carriers, but are convenient to small airports at both ends, GA can save you lots of time.
You will find some difficulty renting aircraft for very long, multi day cross countries. It can be done, but many aircraft renters are not looking to have their aircraft used that way. The main reasons are the extra effort they get into if you have a problem far from home, and the loss of rental opportunity for them, while you have the aircraft away. If you fly four hours a day, they loose four more hours a day (at least) of opportunity to rent that aircraft locally.
You might find a private club who caters to this type of aircraft use, but plan to make an investment. Your best hope is to find someone else (who perhaps owns their plane) who wants to do such a trip, and wants to share costs and flying.
In the long term, I suggest that you save your money to buy even a modest aircraft, it will cost what it's worth, but it will set you free for flying. In the long run, renting is going to cost you a lot more!