Into which directory should I install programs in Linux?

The "more correct" depends on your distribution. You should check your distribution's guidelines on where to put software that isn't managed by the package manager (often /usr/local) OR on how to create your own package for it.

As you said TeamSpeak just put everything in one folder (and may not be easy to reorganise), yes /opt/ is probably best.

(But, for instance, in Archlinux, the package manager can install there, so I'd still make a PKGBUILD to install in /opt.)

Also distributions usually try to follow the Filesystem Hierarchy Standard, so this is where to look for more generic convention.

If you will be compiling your own software then you ultimately control the installation location. By convention, software compiled and installed manually (not through a package manager, e.g apt, yum, pacman) is installed in /usr/local. Some packages (programs) will create a sub-directory within /usr/local to store all of their relevant files in, such as /usr/local/openssl. Other packages will install their necessary files into existing directories such as /usr/local/sbin and /usr/local/etc. These are simply default locations and can be changed during compilation.

When you are compiling software, the installation location can be specified by using the --prefix= option when running ./configure. It is highly recommended that you look at all of the available options for your package by running $ ./configure --help | less. Additionally, browsing the INSTALL and README documents provided with your package is a good idea. They tend to include installation instructions and dependency information that is specific to the package.

It should also be noted that although you can store software anywhere, according to the FHS, source code for locally installed software should be stored in /usr/local/src Standardizing where you store your source trees will allow you to easily locate a tree if you need to copy a stock configuration file or binary. Even though some packages use it, your source code should not be stored in /usr/src as that is designated for system software such as the kernel.

Finally, you need to ensure that your installation location is included in your $PATH. If you decide to install your package in /opt but it's not in your $PATH your shell won't find the executables and you will have to use the absolute path to invoke your programs. Here are some great discussions from AU about configuring your $PATH

Additional reading: man hier

The Linux Standard Base and the Filesystem Hierarchy Standard are arguably the standards of where and how you should install software on a Linux system and would suggest placing software that isn't included in your distribution either in /opt or /usr/local/ or rather subdirectories therein (/opt/<package> /opt/<provider> /usr/local/bin).

Best practice is to convert software you download to a installable software package native to your distribution, for instance dpkg or rpm formats. That will facilitate reporting, upgrading and cleanly removing the software.