Typically, for a solid hardwood floor, you would use a pneumatic flooring nail gun. This would require a compressor as well. Hard to tel from the pictures, but you say these are thin. It’s probably 2 1/4” strips, which is pretty standard in hardwoods. These are tongue and grove, where one end slips into the other. The flooring nailer lines up to place the nail or staple into the tongue of a board, and then you place the next board over that.
Now, people have been installed hardwood floors for a very long time, and fancy nail guns and compressors haven’t alway existed. If you really want to go with this with very basic tools, you will either need to pre-drill through the tongue and hammer in finish nails, and then use a nail set to drive them all the way in. Or, you can face nail (right through the top) with finish nails, nailset them in, and then put some wood putty back on.
With this being a kitchen, 3/4” hardwood is usually installed under your cabinets. Adding 3/4” to your existing floor and butting them up to the toe kick and other appliances can cause issues, especially with dishwashers. They will be damn near impossible to remove if there is a problem.
There will probably also be doors and floor trim to be handled. You really should undercut the door jamb and pull the base molding off before you lay the floor. Hackjob slap it together jobs don’t do this and it looks like shit.
This is all very doable. You have a small space, and given enough time, you can complete this with minimal tools. Basically a handsaw and a hammer and you should, in theory, be able to complete this.
BUT!!!! Because you posted this and would be open to other options, I would HIGHLY recommend you go with a lock and click floor. I much prefer vinyl to laminate. They call it LVT or LVP. Vinyl is easy to cut, easy to install, won’t have the height issues, it’s a floating floor so you don’t need to get them under your cabinets, and you are more likely to get away with caulking and trimming out with shoe molding.
It sounds like you don’t have a lot of DIY or building experience. Start with lock and click flooring, it is geared towards DIY at home type installs.
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