The question I believe can be broken rephrased into the following 2 questions.
- How does one understand the prayers rather than just know what they mean?
- How much understanding is needed when saying the prayers?
The answer to the first question is really very simple. One has to study the tefilot outside of shul and prayer. Learn the history of the tefilot, the changes in tefila from the time of the Mishna till today. Go into the sources of each bracha. One recent book which does this (reportedly well) is "The how and why of Jewish prayer" Reviewed here There are other books on the subject, but this is one I just read about recently which seems very good. Personally, I enjoy going back to mesechet berachot and looking at things there.
The second question is interesting. In mesechet berachot (50a) there is a discussion about Birchat Hamazon. It discussing various nusachot for the zimun, and mentions that people who say the wrong words are boors. It seems here to imply that if you don't say the exact right word, you will have the wrong conceptions of Gd and prayer and will be doing more harm than good. At first glance, both the boor and the correct nusach are saying the same thing, with slightly different grammatical forms, or synonyms. From this discussion it seems clear that the correct general concept is the most important, and the individual meaning of each word is not so important. Meaning, what is a correct nusach and an incorrect nusach is created by the meaning of the words as a whole, rather than the individual meanings of each word. (Both the boor and the talmud chacham say the same indivdual words, but the order they are in and the grammer makes the difference). Since you are reading from a siddur, there is no risk of making a mistake with the words, only with individual translations. So ensuring that you have the correct general concept is more important.
Certainly in chasidic circles, the story of the little boy who could not read hebrew, but could only read the alephbet during Yom kippur services, suggests that the exact words are the least important and the kavnah makes all the difference. (Even if you don't know any words at all to say) Note, this story is not meant to teach halacha, just rather the importance of kavana.