This is the cut list. I'd go on the safe side and get two 1x10x 10 foot pine boards from your favorite big box store. You might be safe with one and a 1x2 for the molding, but having extra material on hand is nice in case of mistakes. Don't worry about getting the clearest pieces, this will be painted. Just be sure you won't need to cut joinery into knots.

I get my pine from a local lumber mill. It's fantastic quality and very inexpensive, but as you can see I have a lot of clean up to do. If you don't want to dimension your own stock, definitely go with the big box store stuff. No planing (except for the bottom) or ripping is required with a 1x10 board. 

The carcass of the box needs 9 1/2" wide boards. Choosing stock wide enough will save time because you won't need to glue up boards. 
If you choose to go with rough boards or a different species, try to get boards at least 10" wide. 

Measuring
You will need the following:
 - Tape measure
 - Ruler
 - Calipers or something to measure thickness
 - Squares. Since these pieces are quite wide, a standard 12" combination square won't quite do. A framing square is a good inexpensive alternative, or do as I did and make your own.
 - Pencils. A marking knife can come in handy too

Sawing (Japanese)
I will be using Japanese saws. These will be the only two required. A Ryoba for cross cutting and ripping, and a Dozuki for cutting the joinery and miters.

Sawing (Western)
A cross cut hand saw is the only big saw needed. There isn't much ripping involved if you use S4S 1x10's. If you anticipate a lot of ripping, than a rip saw would be helpful too. A crosscut carcass saw will be used as well as a dovetail saw. 
If you are planing on using power tools for the ripping and crosscutting, then only a dovetail saw is required.

Planes
I am starting with rough cut stock. With S4S pine from the big box store, I would really only need a No.4 Smoothing plane, or a No5 jack plane. Read the descriptions below to see what planes you'll likely incorporate into your build
 - Block plane: Handy for creating the chamfers on the moldings
 - Smoothing Plane: Finish prep
 - Jack plane: Rough stock removal to bring boards flat and to an even thickness. 
 - Jointer plane: Final flattening. This can be optional here if you cross cut your pieces first. The longest piece is 15" and can be flattened easily with a No5.
 - Router plane: Aids in cutting the dado for the inset tray.

Chisels
Sizes will depend on your dovetail spacing. I like to have a wide chisel on hand to reinforce knife lines when making precision cuts. If I had to choose only two, I'd want a 1/4" and 1/2"
 - A mallet of your choosing
 - A strop. Pine demands the sharpest of edges for clean cuts. If you notice crushing instead of cutting, take the chisel to the strop

Dovetail Layout
This is the kit I use. Yours will vary on the process you use to cut dovetails.
 - Marking gauge: To establish the baselines for the dovetails according to the stocks thickness
 - Small Square: to transfer lines for the layout and to check the floor of the dovetail sockets for square after chiseling
 - Dovetail marker: Optional. A sliding bevel can work here
 - Dividers: These aid in laying out pin spacing. This is also optional as it can be done with measuring tools
 - Marking knife and Pencil: Used to establish and transfer the dovetail layout.
 - Small Ruler: Used to set up the wing dividers. 

Hardware and Installation
My box will differ from the plans. I will be doing something special to the inside of the box and the inside of the box lid, so I won't be using the strap hinges shown in the plans/video. I will be using regular mortise hinges, but probably not these. Something more substantial. 
 - Lock: This is optional and I did not use one on my first box since it sees heavy use in the shop. I will be installing this lock I picked up from my local woodcraft store
 - Nails. brass for the escutcheon plate, and regular finish nails for the bottom and the molding
 - Hammers: I have a nice upholstery hammer for the really small nails and a regular beater for the rest 
 - Nail set: To sink the nails in the molding without marring the surface.

Finishing and Glue
 - Old Fashion Milk Paint in Salem Red. I highly recommend this brand in any color of your choosing. Sean has a great video on the mixing and application of Old Fashion brand milk paint.  If you want the milk paint look but the ease of a latex paint, consider the General Finishes Milk Paint product. I will also be applying boiled linseed oil (not shown) over the paint for added protection and sheen.
 - A decent paint brush
 - Blonde shellac for the inside
 - Liquid hide glue for all of the glued joints. If you haven't worked with liquid hide glue yet, you really should take this opportunity to try it. It's amazing stuff and you may find yourself preferring it in most glue situations.
 - A card scraper for final finish prep. Sandpaper works too and pine is not the easiest wood to scrape, so have some on hand. 
Of course this box would look just as nice with a clear finish if that is what you choose. 

Work Holding and jigs
This is how I do it. Basically, have a way to hold your pieces when you saw and plane them.
 - Holdfasts
 - Winding sticks to check for twist in the rough boards
 - A bench hook
 - A does foot (I do not use vises on my bench)
 - a shooting board (optional)
 - A moxon (twin screw) vise that is not shown here for cutting the dovetail joinery.
 - Brushes. Pine is very soft and any chips or shavings on the bench will leave marks. Be good about sweeping up the bench after each operation and there will be less dent removal later on. A small paint brush is also really good for getting the sticky pine sawdust out of your saws teeth.

Handy to Have
 - A drill of some sorts. I like to drill pilot holes for nails in some cases
 - A thin blade pocket knife. I've found these to be very helpful in cleaning out the inside corners of dovetail sockets
 - An awl to mark locations for hinges, locks, and nails
 - Wax and a rag. Keep your planes and saws waxed and they work so much better
 - A panel gauge. It's like a marking gauge, but for wider panels. This is handy if you need to rip your stock. Unnecessary here if you do not.