I assume you mean the nylon webbing straps. There are a couple good ways to do this! I recommend a practice run or two with the unwanted end so you have a good idea what you're doing. Also, make sure any buckles and snaps are above where you wish to cut. Orienting and rethreading these is not fun.
Scissors and candle
You can easily cut webbing with a pair of heavy duty or fabric scissors. Trim any fuzz or irregularities.
To seal the ends, run them quickly close to the flame of a candle or lighter. Be careful not to burn or overmelt the ends as they will create a big ugly blob of brittle plastic; you just want the strands to fuse together at the very tips.
You can either use a soldering iron / hot knife, an actual knife heated with a torch (please don't use a good one), a pair of heated scissors, or any other hot sharp tool to cut the material, doing both the cut and fuse step in one. The best temperature is slightly above the melting point of the material - any hotter and you tend to get burning. I slightly prefer this method as the pressure of the cut pushes the nylon strands together, helping them fuse.
After cutting and fusing
You should sew the ends after fusing them with a heavy nylon thread. If you have a serger you can use that; otherwise a few passes with a zigzag or locking stitch is fine. Make sure to use a heavy needle. If you are just trying to keep the ends from unraveling or buckles from sliding off, you can fold over the inside twice and stitch over the end as below. You can also make a larger thumb loop if desired, though I've never been thrilled with how they catch in scrub.
Note: if you have canvas webbing, run a couple rows of locking stitches before folding over. Don't try to melt it.
You can purchase metal clips that will "lock" the ends of the webbing. I've used these on canvas webbing before with no issues - search something like "metal webbing ends" or "belt end tips" on amazon for examples. Since you mention being concerned about weight this probably isn't ideal for your situation, but they look great on certain styles of pack.
Also not addressing the weight, but if you occasionally share your backpack with someone who would appreciate the extra length but want to keep everything tidy the rest of the time, safety pins or rubber bands are a very viable non-permanent solution (also there's no such thing as too many safety pins when hiking).