I have a rain water collection barrel attached to a section of my gutter. I have a simple sediment filter attached to the barrel as the water is used inside a studio for a air chiller. I want to increase the pressure of the water so it can flow through a 15' hose (a 50' hose would be even better but not necessary). What pump option should I go with?
It depends on how much height difference more than length of hose. For a couple feet of lift, you can use a fountain pump. But if you need more speed and more lift, then a submersible utility pump will work. Note that the utility pump may not be designed for long term use, they are typically used for draining something that has flooded and then going back into the tool shed. And both expect that the other end is always open.
Depending on exactly how you plan to use this, a sump pump may be ideal to lift the water from a low barrel to a higher one, and then you can tap the higher barrel for your purposes, only opening the tap when needed. You'd want to wire up the pump to automatically cutoff when the top barrel is full.
I agree with the idea of the fountain pump. That's the usual way to go about lifting water to any reasonable height. Beyond the top of your roof line may be unreasonable (O;
Make sure that you are not violating local laws with respect to damming rain water. It can actually be against the law, especially when corporations have purchased certain water "rights."
Most of the pumps you look up have specifications like this based on elevation.
Discharge Flow @ 0 ft. (gallons/hour) 3800
Discharge Flow @ 10 ft. (gallons/hour) 2700
Discharge Flow @ 15 ft. (gallons/hour) 1700
If you have a 50 Gallons Rain barrel it would be empty in minutes. So all you need is a fountain type pump.
The submersible type and => 1/6 Hp are all to big for your job.