I would like to echo the suggestions made by several others that the best way to proceed for a newcomer is by way of "highlight" discs. I first tried it the "hard" way by listening through the entire operas, but got fed up very quickly.
I recently revamped my entire Wagner "highlights" material, and have acquired what I consider to be a decent collection in terms of length and quality. In total it amounts to some 10 hrs 40 mins playing time:
1. Rienzi: Overture plus 'Allmacht'ger Vater', (Rienzi's Prayer) (from Act 5) - Sawallisch/Chor der Bayerischen Staatsoper, Bayerisches Staatsorchester
2. Der Fliegende Holländer (highlights) - Levine/Metropolitan Opera
3. Tannhäuser (highlights) - Sinopoli/ Royal Opera House Covent Garden
4. Lohengrin - Abbado/Vienna Philharmonic Orchestra
*5. Tristan und Isolde - Pappano/Royal Opera House Covent Garden
6. Die Meistersinger von Nürnberg (highlights) - Jochum/Orchester der Deutschen Oper Berlin
7-10. The Ring (separate highlight CDs for each of the 4 operas) - Janowski/Staatskapelle Dresden
11. Parsifal - Barenboim/Berlin Philharmonic Orchestra
What I like especially is that these versions are well recorded with little if any background noise of any sort.
* In the case of Tristan, I think what I did was to select tracks 1, 5, 9, 10, 12, 19, 27, 33 from the whole CD set as I couldn't find a suitable highlights disc that matched the quality of the Pappano recording.