Ok, let's start from scratch then, breaking things down in individual steps.
Part 1: install a new package.
1) download
internettools zip archive.
2) extract the archive to a location where you (read: lazarus) have enough access rights to be able to read the extracted files.
3) start lazarus, and
follow instructions (note the correct paragraph)
This will/should add a new (previously non existing package) internettools to your lazarus list of known packages.
Part 2: Add/install package to your lazarus installation
Now that you've done the previously part, you should be able to
add an already existing package to your lazarus installation.
Part 3; Add the package to the list of dependencies of your project. See Reply #8 made by Mike.Cornflake
... You have to add the package internettools to your project.
Lazarus - Project - Project Inspector - second add button (Add other) - Second Tab Sheet (Add new requirement), from the dropdown list, choose InternetTools.
In addition to the above, it seems that when defining use FLRE, there is also the dependency for this package. I suggested to add this package to Free Pascal, to make it more permanent (and also usable when using stand-alone fpc compiler. It is just a suggestion, not a must).
If you look at the FPC installation inside you lazarus directory, you'll (hopefully) notice a path something similar as:
D:\Apps\Development\Lazarus\fpc\3.0.4\units\i386-win32
the part i386-win32 is also named the architecture. This will be named differently when you are targeting another platform, f.e. (cross) compiling your application for the pi would be using the units located in the path like: D:\Apps\Development\Lazarus\fpc\3.0.4\units\arm-linux
If you have used different build-modes for Free pascal then one way to distinguish is to place them inside another directory that explicitly distinguishes between different builds (in order to prevent constant recompilation when changing build-modes). For example my debug compiled units are located at: D:\Apps\Development\Lazarus\fpc\3.0.4\unitsdbg\arm-linux
That hopefully answers your questions related to architecture and build modes. if not then feel free to ask (*)
(*) In case this is magic to your ears, then please forget about it immediately. Because in that case you do not use different build modes for your FPC units/packages and you should not start worrying about that at this point in time.
Part 4: the FLRE units
1) Inside the archive you've downloaded/extracted is a directory named import/FLRE/src
2) copy these files to a location where you are able to compile them with plain FPC compiler (or use lazarus to open the files and compile them).
3) copy the compiled units into D:\Apps\Development\Lazarus\fpc\3.0.4\units\i386-win32\FLRE (replace the used path in this example with the one used on your system and take note of the architecture that you are using. It would have to match yours).
That should work for you to have a more permanents installation of the FLRE package.
With all these steps you should now be able to open up your project (or start a new one), add the internettools dependency to your project and start using the internettools package. Don't forget to set the internettools parameters correctly before compiling/building your project. e.g. explicitly setting use FLRE define if that is what you wish for the package to use (same as for other options available in the internettools package)
In case of errors, please state which part, which number and what error message you receive and try to post a verbose log (or better yet, try to read the verbose log yourself and see what the compiler actually tries to do). These logs can be very long, so you could cut some stuff away, but take note of parts where it tries to search for the (missing) units.