It looks like no one has given you a clear simple "talk to me like im a 2 year old" answer, so ill give it a try.
There is no good apt like downloader/installer combo for slackware. Things like slapt-get exist but are very....lacking.. and pretty much not worth the time because its easier the other way. Here are your other options:
Method One:
1.
Search for the software you want on
http://www.linuxpackages.net/ and download the .tgz file to your harddrive. The short explanation is that this site hosts pre-compiled software by other slackware users, so you dont have to compile it yourself, which we'll get to next.
2.
After you have downloaded it to your hard drive open up a terminal and "cd" to the directory that you saved the package to. (I like to put all my .tgz files in a single directory /opt/installed packages, but you don't have to do that)
3.
Now on the command line type "installpkg packagename.tgz" this installs the software to your system in its appropriate directories (which can be almost anywhere).
4.
You can now run your software almost always by typing the name of the program or some small variation of it, usually if you type the first 3 letters than hit "tab" it will pop up for you.
5.
To get a list of all the installed packages on your system run "pkgtool", or if you want a nicer looking version install the program Slackins
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Method 2: Compiling from source
1. Almost all the software you want on linux gives away its source code. If you go to their site and find the source code you can then save it to your computer, usually as a compressed file.
2. open that compressed file (there are a lot of ways to do this, look it up or ask) and go into the directory.
3. usually in this directory there will be a file called INSTALL or README which will tell you how to install/compile it. if not you are on your own, below i will outline the typical steps you go through for ~75% of the programs you probly want.
4. type "./configure" (sometimes this step is skipped)
5. type "make" this compiles your software
6. The final step is the most important! You can now probably type "make install" but DONT DO THAT. it installs the software all over your system and you usually loose track of what you did, so instead we want to use a utility called checkinstall.
7. if you dont already have checkinstall, (nothing happens when you type checkinstall on the command line) go to linuxpackages and get it.
8. once you have checkinstall installed when you are done running "make" for the package you were doing before type "checkinstall", almost always the default options are fine, just keep hitting enter until checkinstall prompts you to give a description. At this point i just type in the package name and hit enter twice.
9. Once checkinstall is done your software should be installed, and checkinstall automatically registers the package with pkgtool so it is in your installed package list.
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to remove packages use pkgtool, slackins, or just removepkg packagename