I use Hardie a lot and really like it.
Looking at your picture and having dealt with chimneys that look similar to yours, you really don't know how well the original wood is attached. I've seen a lot of them done with finish nails and roofing nails. They come apart way too easy. I would strip off the existing wood, go over the framing and probably make it a bit stronger. Then wrap it in OSB or plywood.
Wrap it in house wrap and either check the flashing, or install new flashing. Flashing around chimneys is a very common location for leaks into the house!!!
To cut the Hardie, you need a special Hardie blade for your circular saw. I have several of them, including one for my cordless saw. They all work great, but the more expensive ones last a lot longer.
Do the corners first. I prefer to use 3 inch deck screws to attach the trim. It takes a second or two for the screw to work its way into the trim, but then it goes in just like wood. I also use star drive screws and a cordless makita impact driver.
To install the lap siding, you can either use roofing nails in a coil nail gun, or hit them in by hand with a hammer. You can also use screws. Hardie makes their own line of screws designed for their tile backer board that work great on everything. They are heavy duty, coated and have a square drive. It's your call, but roofing nails are the most common way to do it. Just be sure to snug up the nail onto the Hardie siding. You never want to crack or damage the Hardie. The head of the nail should just touch the Hardie. With a hammer, this is easy. With a nail gun, it takes a little adjusting to get the setting right. Even then, some will go in too far, others will need an additional tap with a hammer to get them in all the way.
Nail at the top of each piece of siding and then cover the nails with the next piece of siding as you go up. Use good 40 year caulking to cover your top row of nails and your screws. Caulk down the sides of the trim and then use a good quality paint. Once painted, you will never have to touch it again. The only brands of paint that I like are Valspar and Sherwin Williams.
