I'd agree with /u/vythurthi's suggestion of Lohengrin, which is very similar in style to Tannhauser. I'd also suggest The Flying Dutchman, which is closer to Meyerbeer and grand opera style, while still being distinctly Wagnerian. Since you like Strauss, I'd suggest listening to some of his operas A) because they're really the central part of his output and B) because he takes a lot of Wagner's concepts and elaborates them extensively. I tend to recommend Salome because I think you get the best of both worlds, whereas Elektra is a bit extreme for beginners and Rosenkavalier can be a bit sickly sweet if you're unused to Strauss.
I think /u/Shyguy10101 is right to recommend Mozart and Rossini too. However, while Wagner did take quite a lot of lessons from Mozart, they do belong to a quite different style and era. They are are a lot more accessible, but depending on your preferences, they may not necessarily appeal to you. The Magic Flute (while apparently silly) is actually one of Mozart's most "advanced" works - there are scenes where he experimented with melodram which anticipates a lot of what Wagner was to do decades later. You can also follow that tradition back a bit further to Gluck (who Wagner admired to the point of creating a German version of one of his French operas), but he can be hard to get in to. Orfeo ed Euridice is acccesible and short though.
In between those two points in the German tradition you also have Wagner's big influences like Beethoven's Fidelio (you can also listen to the longer early version, Leonore, if you prefer) and Weber's Der Freischutz. Wagner was also very influenced by the scale and use of repeated motifs in the work of Berlioz, although these are generally less accessible. If you feel adventurous, you could try Les Troyens (long but underrated) and Roméo et Juliette (a bit of a mixed bag and not strictly an opera, but very influential). La Damnation de Faust is also good and very close in date to Tannhauser, but if you ask me it's a lot more... French. Somehow. If you feel like exploring some "Grand Opera" (another big influence on Wagner, then Les Huguenots is a good introduction.
If you like the lush late romantics, then there's also Puccini, where there's a big Wagner influence combined with the lyricism of the Italian tradition - so again, you get the best of both worlds. Tosca is probably a good place to start - great story, massive set pieces, feels like a film as much as an opera, etc.
Taking a chronological approach, you could try some stuff by Wagner's great rival Verdi from around the same time, like Nabucco or Rigoletto.
Of course you could just do what I did when I first listened to opera properly, and dive straight into the ring cycle! It's time consuming but well worth it, and once you've worked through that you're over the biggest hurdle and you could end up completely hooked.