So first off how to find/catch your jumping spider
They can be found almost anywhere. On the side of your house, gardens, in the grass, on trees, and possibly inside your house. All you have to do is look around a bit. It may take you a couple hours to a couple weeks to find one. You should probably look up which types of jumping spiders live in your area so you can easily identify them.
You'll need the container that you'll be keeping your jumping spider in.
A piece of paper.
And your soon-to-be pet jumping spider
Set down the container on its side and very gently coax the jumping spider into it. It will most likely want to get out at first but you can just put the top on the container.
DO NOT TRY TO GRAB OR PINCH YOUR JUMPING SPIDER THEY COULD GET HURT AND THEY WILL BITE
Next is where you will be keeping your jumping spider
They need space so don't put them in a little tiny jar, put them in something bigger but not to big like a huge aquarium. A 12 quart (32 ounce) Mason jar is about the right size for your jumping spider. Don't put on the top of the mason jar!! It won't be able to get air and it will suffocate!! Get a paper towel and a rubber band. Put the paper towel over the mouth of the jar and put the rubber band around the mouth of the jar holding down the paper towel. Poke some small holes (Not big enough for the jumping spider to climb through) in the paper towel with the tip of a pencil. It should be able to breath well like that.
They need to be somewhere in your house where they can get some sun. Be careful not to leave them in the sun too long or they will burn, glass works like an oven. In order for them to stay healthy they have to warm their muscles in the sun and run and jump. If they do not do this they will get weak and die. You can put their container outside, partially in the shade of course, on sunny days.
They also need to have some sort of an area where they can make a sleeping pouch (it looks kind of like a cocoon and yes jumping spiders can make silk too, just like any other spider). It can be a curled up leaf, napkin folds, or something else they can get under or inside of.
While I'm talking about sleeping pouches I should probably mention cleaning. You should clean their container twice a week and get rid of things like insect bodies, droppings, and stray bits of web.
You should also try to keep your jumping spider somewhere where there won't be a heck of a lot of noise, movement, or sudden changes in environment as to not startle or scare your jumping spider.
Listen, if you can't catch a fly or cricket and get it in the cage without killing it or having it get away, you are not suited for keeping any carnivorous bug as a pet. Anyways... Try to feed your jumping spider every other day. They should never go a week or longer without eating. If you feel as if you are unable to catch enough food for your jumping spider you need to let it go outside. Don't feed them anything that can sting, or really inject venom of any type, has a hard shell (beetles and rolliepollies), or is much bigger than your jumping spider. If you have a very small jumping spider and can't find small insects to feed it once again you need to let it go. And like I said with my first note they all have preferences.
Now for handling
You should probably let your jumping spider live in it's new home for about a week without to much interactions besides feeding it and cleaning it's cage. Don't feel obligated to hold your jumping spider. It isn't required to play with them like with other pets such as dogs and cats. Most jumping spiders are better off not being held and are happy just staring at you from the inside of their cage. When handling them you should make sure you wash your hands before and after you handle them (Wash with plain water. Soap could be too strong and hurt them) and dry your hands well. They can drown in any amount of water because of how they breath through their abdomens with 'book lungs'. If any water gets between the 'pages' of their lungs they will drown.
Always be extremely gentle when holding your jumping spider. If your jumping spider doesn't want to get on your hand, don't force it. They will be more comfortable with you in time. Some jumping spiders do not want to be picked up or interacted with very much, you should leave them be instead of annoying them until they finally bite you.
All insects and arachnids molt. Molting is pretty much when a juvenile insect or arachnid grows out of its exoskeleton and then sheds it. Most of the time an insect or arachnid will molt while hanging upside down on a leaf or branch. After your jumping spider molts it needs a couple of days to harden and get used to its new exoskeleton. Do not touch it or put food in its cage during this process or you will kill it. Most juvenile insects and arachnids look much different than the adults of the same species. In praying mantises they don't look much different than adults and are called nymphs. Young jumping spiders look very similar to adults besides some color variations.
Jumping spiders live for a year or two depending on how well you take care of them so be ready to make trips to the pet store in the winter. There are a ton of different species of jumping spiders, some of the most common ones in my area are phidippus audax and some of the platycryptus. Look up the ones in your area to identify them easier. Breeding jumping spiders is some crazy stuff that I haven't really wanted to deal with yet so you're going to have to go find info on that somewhere else. Telling the sex of jumping spiders depends completely on the species. There is no fool proof way to tell a jumping spiders sex aight.