Try watching anime subtitled (see how much you can pick up on just by doing that)
Once you get proficient enough, you can take it up a notch and just get rid of the subtitles to see how much you can understand without a translation. I think at about 3 years of college classes (according to an internet guy by the name of GaijinPunch, at least) and about a thousand of vocab words would be a good point to be searching for anime without subs. Oh, and don't forget to use a dictionary for words you don't know while watching. I'd rather you enjoy the show, then note down the words you don't know if you still remember them and study off those for a while. It works for me, at least.
I hate reading subtitle while listening to anything japanese because it confuse me so much. The structure of the japanese language seem to be really different from the English language.
It all depends on preference. Personally, I agree with you on this notion because it seems really difficult for me to technically multi-task between hearing Japanese audio and reading English text, all at fast speeds and rapid succession, too. Some people can pull it off, but IMHO, it more often than not hinders my enjoyment of the content I'm trying to watch.
Also, yes, the Japanese sentence structure takes some getting used to, but the key thing, according to Tae Kim, is that you can have a functioning sentence with just a single verb. The verb is usually the crux of the Japanese sentence structure, considering that almost every Japanese sentence usually ends with one. It shouldn't really take too long to get used to it, but I can't really speak for everyone when it comes to how fast certain people learn things.
I'd like to suggest that you try an anime that is not one of your action-oriented or mystery ones, but something a little slower (Kaasan Mom's Life or Azumanga Daioh) and more like every day life. You might pick up more common day phrases. I was in Okinawa a month ago and found myself struggling with asking how much something was in a grocery store, or what the item was, where to find a shop or particular item in the shop, etc.
Also, I tried learning specific phrases for ordering lunch at my favorite Japanese restaurant near my office in SF and the owner got a kick out of my ordering.