First thing, most branches are very poor for turning into lumber. They will be stress wood and very likely to bend and twist as they dry or after, sometimes even while you are cutting them.
They of course can be used for turning.
As Peter Grace said, always ask before you collect it, but if it's just laying around, most of the time you will be allowed, some might be very grateful to get rid of it.
Now for processing and storing for later. You have many options, but if all you have is a table saw you are asking to loose parts of your fingers.
If you are saving the larger pieces for turning, splitting it in half and drying it like firewood is a good idea.
Peter also recommended the chainsaw mill. They can work and they are a LOT of work, but likely the cheapest option. Not only do you need to buy the guide, you need (should) to have a special chain for the saw. Most chains are designed for cross cutting wood (across the grain) and to make lumber you should have a ripping chain (along the grain), cutting slabs like that is not only hard on the person it is hard on the chain saw, and not having the correct chain is MUCH harder on both.
Also be aware that unless you are an expert sawyer you are not going to cut nice flat cuts. so you need to cut them larger to be able to plane down to your desired thickness. You also need a guide to make your first cut 'flat' and not twist around the log.
Now, you can make lumber this way and MANY people do, the equipment is relatively cheap and after a dozen logs your will likely be doing a reasonable job.
Some people also use bandsaws to make lumber, getting a large throat like 12" would be my minimum recommendation and then you would also need to build some guides to help run the logs through.
Band saws will waste a lot less wood because of a much smaller kerf, chain saws are going to be at least a 1/4" for each cut and the more uneven the cut you more you have to waste planing it. Bandsaw kerf is about 1/16" or less.
You can also find in many communities people who have portable mills and can be paid to cut up the lumber.
Last you can get your own, My dad and I bought a Woodmizer LT15 (used) 8 years or so ago and make our own. It's been worth every penny.
However, with all of these options you still have one thing to worry about, often trees that are 'harvested' in town have many things inside that are dangerous to blades of any stripe. Nails, barbed wire, even large parts of bicycles.
Once you have it cut into lumber, you need to 'sticker' it and let it dry for quite a while, species and environment dependent for how long.
You can also pay to have it kiln dried or make your own small solar kiln.