I'm not sure this is a winter project unless you already have a lot of cold air blowing in underneath. Start by removing the old skirting and buy new skirting the same height(if the old stuff was sized right). Most skirting has a U shaped channel/track on the ground that is held down in the ground with gutter nails spaced every few feet. The bottom edge of the mobile home has a 6" or so channel/panel screwed into the side along the bottom which has a bend in it to allow space for the skirting to slide into. It then installs much like a sliding glass door where you slide the skirt up into the top panel and then down into the U channel. The skirting wants to be somewhere around the middle or a couple inches from the top of the panel so the skirting will slide when ground heave moves the trailer or skirting up and down.
Depending upon the type of skirting, each panel may interlock with the one next to it. Some of it like corrugated tin may be screwed. Finally, there may be screws in the bottom channel (or top which defeats the ability for it to float) to hold the skirt in. The skirting needs to float in the top channel and not be screwed in or it won't float and then will buckle with ground heave.
The last one I reskirted I found it easiest to just remove the top channel. It was held with hex head screws and it was all removed in ten minutes with a power drill. Then remove the skirting and the track. Since the old skirting and track has been bent up and mangled from mowers, etc. it isn't probably saveable. Use a hoe to relevel the ground and install the bottom channel. Then install the skirting and top channel one section at a time till done. Make sure the corners are squared up good and start from a corner.
Don't forget access panels where electrical and plumbing and crawl space access areas are needed. If the skirting interlocks for the length of 60' to 70' then it is near impossible to remove a panel later for access depending upon how the skirting interconnects. If you are using some of this stuff then stop at the access places and just use a panel of skirting that overlaps each side by a few inches and is then screwed into the panel to hold it. You can also cut the top channel with tin snips to allow bending of the top channel to remove the access panel if necessary.