Jack Bradley has recorded several conversations entirely in Klingon, see jatlh DeSDu'.
There is an audio-video course entitled "Learn Klingon the easy(ish) way" by ViaLingo.
Grammar
Much detailed information is found in the KLI's HolQeD, but that's usually more advanced level grammar so not suitable for beginners.
Browsing this wiki will answer a lot of questions as well. A good starting pint is the page How do I say...?.
Vocabulary
A good way to memorize words - for some people - is to color the words in your book according to word type. See Coloring words.
If you use an android phone, get the dictionary app boQwI'.
Join Memrise and start learning the words in the KLCP one list.
Getting in contact
If you like to learn and use Klingon in personal contact, consider visiting a qep'a' in the USA or the qepHom in Germany, or try to find out if there is a qepHom nearby your area. Note: You don't need to be an expert to visit such an event.