One thing many members have not mentioned to address one aspect of the OP: As a new collector, he/she would benefit better if we can share our approach/method.
As some mentioned, when they first started out, they put a lot of things in 2x2 that latter on deemed not worthy of the effort(like Carl said, put a 2cents worth coin in a 2x2 that cost 4cents). I did similar thing when I first started out.
Considering the hassle of taking the coin out, once put in 2x2 (either to reuse the 2x2 or any other reason).... it take less than 10secs to put the coin in a 2x2, staple, and put in the box but it take about twice that time to remove the coin (while retain the usability of the 2x2).
To address this dilemma, I bought a few hundred 2x2 ziplock bag (cost about the same or less than 2x2 flip on
Amazon) and throw coins that I am not sure if I should put in 2x2 or not (or those coins that are not ready to be catalog/organize). Doing this still protect the coins from contact mark/scratch/environment and at the same time make it easier to take them out + save on the staple. NOTE: This is temporary/short term storage.
The second approach to address the above dilemma is instead of put the coin in 2x2 ziplock bag, I put the coins in 2x2 plastic flip. This intermediate term storage is for coins of higher value ($10 and above), especially if I know can't come around less than 6months to catalog them.
When consider putting the coins into 2x2 paper flip, beside reasons others had suggested (as to why a coin should be in a 2x2, such as for protection or you want to save/keep); I personally set some requirements for my coins to be in 2x2:
a) Value on the coin at around $5 or more; if btw $1 and $5 then consider additional criteria
b) Uncirculated or PF
c) Ready for catalog/label/storage
Some other aspect to consider: Invest some money in a flat-clint stapler (about $10 - $20 at coin show or online for a small one) and buy flat-clint staple (the standard staple bulge on the back, which would take up more space & can scratch other coins/2x2). Otherwise, you will need a pliers and flatten those staple's bulges (time and efforts required).
One other thing that the OP mentioned: Running out of staple. Other have suggested using masking tape, I've seen people only staple once (on the opening side of the 2x2 flip, when clap together), or twice (at the two corner of the opening).
I use flat clint staple for mini staples by MAX (No.10-5M, box of 5000), each strip have 100 staples. I staple 3-times on my 2x2 (for all coins in quarter size or smaller), once at the opening, and the other two at about 180-degree from the initial one, making sure the staples are tangential to the coin's curvature. In the past couple months, I've save about half a box of staple by this approach and spent about 5 seconds to staple. NOTE: IF you want to keep as much air out as possible or the coins are dollar size, you should do 4 (3 at the 3 opening + 1 at the remaining side) to also keep the coin from sliding.
As always, each of us have our own style/preferences, you can have your own or adopt a little bit from each members on here. Good luck and have fun.