You should take a look at this Wikia page about options to compile LaTeX from Vim. The vim-latex plugin should cover all your needs (e.g. Multiple file LaTeX projects), you can find the full documentation here or with
:help latex-suite from within Vim.
An alternative method is to tweak the
&makeprg option in order to change the program used by the
:make command. From the same article; if using
pdflatex, add this to
let b:tex_flavor = 'pdflatex'
set makeprg=pdflatex\ \-file\-line\-error\ \-interaction=nonstopmode
set errorformat=%f:%l:\ %m
:h :_% and
:h :_# for their special meaning in
&errorformat is used to describe the output, available in the
quickfix window after each run. Here is how those are "dynamically" defined in the plugin for LaTeX-suite.
Once you are able to execute
:make successfully, just add a mapping to save, compile and open (this assumes the file is created with the same location and name, and that
open is an actual PDF viewer in your PATH):
:nnoremap \lv :w<CR>:make<CR>:!open expand('%:r').pdf<CR>