There is no other way than learning step by step and by keeping the common goal in mind. It is just like with reading a book, when you are young and just learning to read, you stutter at letters first, then words, then sentences. In the end you are able to read quickly and recognize words and patterns instead of actually reading every letter. The bset emaxlpe of tihs is the ailbity to qukcliy read wrdos with only the first and last letter being in the correct order.
As a reasonable sightreader i often catch myself doing this in some form as well with sheet music. For example starting a left hand accompanying arpeggio on the wrong note of the chord. This is because i recognize the pattern (a certain arpeggiated chord), but do not read every note separately.
I would suggest starting to learn (parts of) pieces in all keys with not that many modulations, this helps you to recognize the patterns for the base chords in a key. Slowly look for music with more modulations (and therefore, sharps, accidentals etc). Keep in mind that with tonal music, every accidental, sharp or w/e is usually a route to a logical destination and therefore should not come unexpected. You know this already by ear, now you just need to learn how it looks like on paper.
At your starting evel, i would suggest pieces like Mendelssohn's Duetto (from lieder ohne worte), which has nicely arpeggiated chords, slow modulations etc. Other interesting pieces could be Chopin's prelude in C minor or Grieg's Ase's death, both more chord based without arpeggio's. If you start off slow with them, they should not come that hard.
One last thing to keep in mind is that you will need to be able to play without your eyes on your fingers, this is the second important thing besides the reading part. However this you can practise by just playing the pieces you know with your eyes closed.
Goodluck!
Hans