I have noticed while monitoring/attempting to answer common jQuery questions, that there are certain practices using javascript, instead of jQuery, that actually enable you
There's already an accepted answer but I believe no answer typed directly here can be comprehensive in its list of native javascript methods/attributes that has practically guaranteed cross-browser support. For that may I redirect you to quirksmode:
http://www.quirksmode.org/compatibility.html
It is perhaps the most comprehensive list of what works and what doesn't work on what browser anywhere. Pay particular attention to the DOM section. It is a lot to read but the point is not to read it all but to use it as a reference.
When I started seriously writing web apps I printed out all the DOM tables and hung them on the wall so that I know at a glance what is safe to use and what requires hacks. These days I just google something like quirksmode parentNode compatibility when I have doubts.
Like anything else, judgement is mostly a matter of experience. I wouldn't really recommend you to read the entire site and memorize all the issues to figure out when to use jQuery and when to use plain JS. Just be aware of the list. It's easy enough to search. With time you will develop an instinct of when plain JS is preferable.
PS: PPK (the author of the site) also has a very nice book that I do recommend reading