With nine acres and part time or casual burning in the wood stove, you should have an endless supply of wood. The time to cut wood is whenever you are ready. There may be some advantages to cutting when the sap is down in the roots, i don't know for sure, but it is always better to get the wood cut, split, and stacked sooner than later. Start cutting wood before you put in the insert - seasoned firewood takes longer to get than a fireplace insert. Winter is a good time to start, as long as snow isn't too deep.
The basic equipment you'll need is a chainsaw and a maul (plus protective gear - goggles, ear protection, gloves, etc.). I'd start with a down tree, and practice cutting it to the right length. you'll get your saw stuck a few times, but that it the only way to learn. As you cut any log, think about which way the log will go when your cut starts to weaken it too much, and you'll be on your way to avoiding having the saw pinched.
Some guys cut up all the dead wood they find, but I like to leave some dead wood. Dead standing trees are great for wildlife, so I leave some of those, and downed wood is also important, plus it is often rotten. I'd cut some of the fresher standing dead wood, especially oaks hich seem so solid that they aren't the best for wildlife (fewer holes than softer trees). Eventually you'll want to cut live trees. I select the trees to cut based on what I want to leave alive. I choose the nicest trees that will grow into fine specimens and decide what should be removed to give them room. Among the trees I want to remove, i select those that look easy to fell and take them first. Look for a smaller tree that leans slightly toward a clear spot for your first one. Read up on how to fell, and be very careful while doing it. A live instructor is great, so maybe you know someone with some experience. Practice on trees that look too small to kill you until you are very confident.
I wouldn't worry too much about the type of wood, except to understand how long it takes to season. Any wood you have will burn greta as long as it is seasoned. Softwood and softer hardwoods will be ready to burn next winter if you cut, split, and stack them this winter. Oak, hickory, sugar maple will need an additional year. Most people burn about three cords per year if they are semi-serious about keeping a fire lit.