I am mostly self-taught. Here's an approach that has worked pretty well for me. Note that most of this is letting your language muscles and your intuition work on the text, not your analytical reasoning. Note also that only Step 4 involves thinking in metalanguage, and the other steps involve pushing language through your eyes, ears, and mouth.
- Pick a section that is at least 7-10 sentences at a time. As you get better, increase the amount gradually to a chapter.
- Read the section, preferably out loud.
- Listen to the section 2-3 times, or read it out loud 2-3 times.
- Use Alpheios to read the section, looking up any words or forms you do not recognize, using the links to inflection tables and the Smyth grammar that Alpheios provides. (I imagine Logos or other software works very well for this, I do not have them.)
- Listen to the section 1-2 times, or read it out loud 1-2 times, to get the intuition for what you have just worked through analytically in step 4.
If you have loads of time on your hands (I usually do not), once you have done all that, it is time to pull out some of the older commentaries that comment on the text in detail, or to post questions on B-Greek for things you do not understand. Don't do that until you have worked through the text intuitively, as described above.
I think this works really well to help develop reading intuition. It doesn't teach you how to write Greek or speak Greek. I suspect you have to write or speak Greek to do that, and I suspect that is harder to do without a teacher. If you have access to a teacher, or are willing to move to Minnesota, Israel, or Malawi, go for it!