Pre-build coops and/or coop kits seem to be very expensive compared to buying the material to build it yourself, so you seem to be ahead of the curve there, but the caveat to that is you have to have the right tools and skills to build one yourself. You don't sound quite as confident in this area.
You specifically mentioned buying wood so I will try to help with that as best as I can. Since I don't know your level of skill, knowledge or experience with carpentry, I will apologize ahead if I am making this too basic, but I just want to make sure I'm being thorough.
1x3, 1x4, 2x3 and 2x4 usually come in lengths of 8, 10, 12 and 16 feet long. 2x3 and 2x4 are for framing (the skeleton or support part of the structure) and the 1x3 and 1x4 are for cosmetic trim pieces and can be used to form lightweight doors and windows.
While the dimension 2x4 is in inches, the wood does not actually measure 2 inches by 4 inches. It was this size before it was dried in a kiln. The drying process causes it to shrink so 2x4 is the nominal size. The actual size will be about 1.5 x 3.5 inches but it's easier to say 2x4. The lengths (in feet) will be exact.
The most common types of wood will be soft woods like yellow pine or douglas fur. These will be fine for what you need. Other woods from hardwood trees will be more expensive and you do not need the extra strength they provide and you are not making furniture so stick to the common softwood lumber.
You will have the option of standard plain ol' lumber and pressure treated lumber. If you go to a lumber yard and ask for "ten 2x4s in 8ft lengths" the first question they will ask is "Treated or non-treated?" Because they sell both. Once the wood is kiln dried into lumber, it can be sold as is or pressure treated, which is a process of soaking the wood in a chemical while inside a big pressure cooker. This wood will have a green color to it and be much heavier due to the excess liquid content. Most times the liquid will get on your hands when you handle the wood so don't wear white clothing. The chemical prevents the wood from rotting and deters termites and other insects from trying to burrow into or eat the wood.
Pressure treated wood is used where you have wood in contact with the ground, or exposed to the elements. Non treated wood is used inside where it will not be exposed. Keep in mind that 2x4s used to make roof trusses and wall frames can be considered inside when they will be covered by other layers of materials (roof shingles, sheathing, siding, etc) that keep them dry. Pressure treated lumber costs more due to the extra processing involved. Pressure treated lumber can and should be painted, sealed or stained but you have to wait a month or two for the chemical to dry first. The chemical protects the wood from rotting from the inside out. The paint/stain/sealing finish is to protect it from the outside.
Note there is a difference between "ground contact" and "buried." lumber that is marked or sold as treated for ground contact should not be buried. There is lumber marked specifically for being buried or underground use.
Last, there are number grades assigned to lumber. Most commonly you'll see #1 and #2 lumber. This is a measure of how straight the board is in each plane, whether there are knots, gouges or other physical defects in the wood. #1 is the best quality and will cost a just a little more. The boards should not be bent, twisted, curved warped, should have no holes where knots were or any other phyiscal issues. #2 will have more of these issues and will cost less. Depending on what the user will do with the wood, it may be more economical to buy and work with #2 lumber. Lowes and Home Depot usually carry #2 quality wood so you have to look at each piece carefully when selecting it yourself. Sometimes a piece with a damaged end can still be used if you know you will be cutting that end off. I found that the box stores are hit or miss with useable wood. If they recently got a delivery of it in, then there is better wood to choose from. If not, then all that will be left is heavily damaged, bent, twisted pieces that everyone else passed on. Lumber yards usually sell #1 quality and their #2 quality lumber is typically much better than box store #2 quality wood. For me, I live close to a lumber yard so the difference in price of their #1 and box store #2 is offset by the cost of driving all the way to the box store and back.
The wall sections are usually framed with 2x3s or 2x4s and then covered with some kind of plywood or other corrugated metal sheets. Plywood most commonly comes in sheets that are normally in the following sizes (all lengths here are in feet) 4x8, 4x4, 2x4. Some lumber yards and lumber sections of stores like Home Depot will cut a 4x8 sheet down to a smaller size for you even though you will still have to buy the entire sheet. The thickness is measured in fractions at 32nd of an inch intervals like 25/32" and can be pressure treated as well. They come in A LOT of different thicknesses.
To keep this from getting any longer, I will post a link to a Home Depot page
https://www.homedepot.com/c/ab/types-of-plywood/9ba683603be9fa5395fab909d37f448 with a good description of plywoods, how they are made and what the different ones are used for. For chicken coops, they are basically used as the floor and walls, so select something that can stand up to repeated exposure to the weather and chicken droppings. Plywood can available treated or non-treated and can also be painted, stained and sealed for extra protection.
With nearly all construction projects, you want to budget for a minimum of 15% more material than the plans call for. This accounts for mistakes and waste material. If the plans call for a 6 foot piece of 2x4, you will need to take an 8ft piece and cut off 2 feet. If that little 2ft piece won't be used anywhere else in the project, it's waste. This will inevitable happen so don't try to plan on using every inch of every piece of wood you buy.