How many LED Strips can we connect to an Arduino Uno? Is there a way to connect 1-3 meters worth without needing extra parts (relays)? I am looking at Adafruit to purchase some LED strips for a mod for my computer case, and want to power it with an Arduino.

I am looking at the Uno, but the Mega looks good too.  I hear the Uno is good because if the chip fries, you can just replace it easier than the Mega's chip.  I did hear that the Uno chip had issues, but that was a 2013 post.

Anyways, I am essentially looking to connect about a foot of 144 leds/m to the front of my case, and connect that to the Arduino.  I will then be lining the inside of my case with about 12-15" per side (x4 sides) of 60 leds/m so I wanted to connect that to another part of the Arduino.

I read that each led takes something like 6mA or 60mA each, so that is what we should watch for.  I was trying to search for this question that others asked, but people were looking to connect a lot of strips, and needed relays and whatnot, so I wanted to make sure the Arduino itself could power about 1.5 to 2 meters of Led strips in at least 2 different strip configs (or whatever you want to call them).

I also am curious if there will be an issue programming each strip setup or, if it just is a little more work to get it setup?

Since this is a computer cause led setup, I'm not sure if I should go USB power, or a wall charger?  What is the recommended approach for something like this?

I will definitely be reading up more tutorials and all that important stuff, but I want to buy the stuff now since I realized there's a cyber Monday deal going on, so I don't want to miss out...

Any help is appreciated, thank you!
Yes, the current draw is the main factor here.  That's not really an Arduino issue though, as I will enlighten you about later on when I talk power supplies.

I assume, since you don't mention what the LEDs are, that you are talking about strips of WS2812 "NeoPixel" LEDs (the ones where you can control the colour of each and every individual LED).

I would recommend keeping all the LEDs as one single strip.  That doesn't mean you can cut the strip into segments, just join the ends to each other using wire in one chain.  That way you just have one chain to deal with in the Arduino.  Running multiple chains can be somewhat more resource hungry and I'm not even sure it's possible on an Arduino.  So keep it all as one long chain split into chunks and re-joined with wire.

Now for power.  Your computer has a big powerful power supply in it.  That gives a very good regulated 5V supply with many many amps available.  More than enough to run hundreds of LEDs.  By all means run the Arduino from USB (makes for easy programming - inspect your motherboard, it may well have an internal USB connector that would be ideal (I know mine has).  And then run the LEDs themselves off the +5V direct from the computer's power supply.  A spare hard drive power connector could be good for that.