MikeWells:
So from my understanding, I should first choose an LED strip to use then build from there?
Pretty much.
Well, step zero is to get your specifications complete. How long a strip, what colours and patterns, what brightness. The first two you have mentioned, the third not. You're probably best off with an RGB or RGBW strip (the latter has better whites but of course costs more).
For evening decoration of a home, a 30 LED/m strip will do just fine, but it won't be very visible in the daytime. If you need that, get a higher LED density for more brightness - you can always tone it down in the evening.
Will it be exposed to the weather? If so, waterproofing is needed.
Strips come in 12V and 24V - of course that makes a difference. A 24V strip has half the current of a 12V strip so thinner wires needed, especially a concern when the strip becomes long. Lower currents are easier to handle, but higher voltages less so, and faults like short circuits are more damaging at higher voltage.
So now you have your strips - that tells you the voltage and current you have to control, and how many channels of this. The next step is figuring out how to control them: most likely a MOSFET circuit for each strip will do fine. Most likely a regular logic level MOSFET like the IRLZ44N or IRL540 will do just fine for this. Those can be controlled easily by an Arduino.
For power, maybe easiest is two power supplies. A 12V or 24V for the LED strips (make sure it can supply sufficient current), a 5V one (mobile phone charger, USB adapter) for the Arduino. Do share grounds, or it won't work. Just a single ground connection between the two.
As you now have your hardware design, it's time to think the software. For a strip that's just fading one colour into another and back again that will be pretty simple.