Where to place $PATH variable assertions in zsh?

Here is the docs from the zsh man pages under STARTUP/SHUTDOWN FILES section.

   Commands  are  first  read from /etc/zshenv this cannot be overridden.
   Subsequent behaviour is modified by the RCS and GLOBAL_RCS options; the
   former  affects all startup files, while the second only affects global
   startup files (those shown here with an path starting with  a  /).   If
   one  of  the  options  is  unset  at  any point, any subsequent startup
   file(s) of the corresponding type will not be read.  It is also  possi-
   ble  for  a  file  in  $ZDOTDIR  to  re-enable GLOBAL_RCS. Both RCS and
   GLOBAL_RCS are set by default.

   Commands are then read from $ZDOTDIR/.zshenv.  If the shell is a  login
   shell,  commands  are  read from /etc/zprofile and then $ZDOTDIR/.zpro-
   file.  Then, if the  shell  is  interactive,  commands  are  read  from
   /etc/zshrc  and then $ZDOTDIR/.zshrc.  Finally, if the shell is a login
   shell, /etc/zlogin and $ZDOTDIR/.zlogin are read.

From this we can see the order files are read is:

/etc/zshenv    # Read for every shell
~/.zshenv      # Read for every shell except ones started with -f
/etc/zprofile  # Global config for login shells, read before zshrc
~/.zprofile    # User config for login shells
/etc/zshrc     # Global config for interactive shells
~/.zshrc       # User config for interactive shells
/etc/zlogin    # Global config for login shells, read after zshrc
~/.zlogin      # User config for login shells
~/.zlogout     # User config for login shells, read upon logout
/etc/zlogout   # Global config for login shells, read after user logout file

You can get more information here.

tl;dr version: use ~/.zshrc

And read the man page to understand the differences between:

~/.zshrc, ~/.zshenv and ~/.zprofile.

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Regarding my comment

In my comment attached to the answer kev gave, I said:

This seems to be incorrect - /etc/profile isn't listed in any zsh documentation I can find.

This turns out to be partially incorrect: /etc/profile may be sourced by zsh. However, this only occurs if zsh is "invoked as sh or ksh"; in these compatibility modes:

The usual zsh startup/shutdown scripts are not executed. Login shells source /etc/profile followed by $HOME/.profile. If the ENV environment variable is set on invocation, $ENV is sourced after the profile scripts. The value of ENV is subjected to parameter expansion, command substitution, and arithmetic expansion before being interpreted as a pathname. [man zshall, "Compatibility"].

The ArchWiki ZSH link says:

At login, Zsh sources the following files in this order:
/etc/profile
This file is sourced by all Bourne-compatible shells upon login

This implys that /etc/profile is always read by zsh at login - I haven't got any experience with the Arch Linux project; the wiki may be correct for that distribution, but it is not generally correct. The information is incorrect compared to the zsh manual pages, and doesn't seem to apply to zsh on OS X (paths in $PATH set in /etc/profile do not make it to my zsh sessions).

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To address the question:

where exactly should I be placing my rvm, python, node etc additions to my $PATH?

Generally, I would export my $PATH from ~/.zshrc, but it's worth having a read of the zshall man page, specifically the "STARTUP/SHUTDOWN FILES" section - ~/.zshrc is read for interactive shells, which may or may not suit your needs - if you want the $PATH for every zsh shell invoked by you (both interactive and not, both login and not, etc), then ~/.zshenv is a better option.

Is there a specific file I should be using (i.e. .zshenv which does not currently exist in my installation), one of the ones I am currently using, or does it even matter?

There's a bunch of files read on startup (check the linked man pages), and there's a reason for that - each file has it's particular place (settings for every user, settings for user-specific, settings for login shells, settings for every shell, etc).
Don't worry about ~/.zshenv not existing - if you need it, make it, and it will be read.

.bashrc and .bash_profile are not read by zsh, unless you explicitly source them from ~/.zshrc or similar; the syntax between bash and zsh is not always compatible. Both .bashrc and .bash_profile are designed for bash settings, not zsh settings.

I had similar problem (in bash terminal command was working correctly but zsh showed command not found error)

Solution:

just paste whatever you were earlier pasting in ~/.bashrc to:

~/.zshrc