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)?
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.