Parroting what’s already been mentioned, repeated exposure and context both help with this.
The manga series よつばと！ could be helpful for this because (after the first volume) the titular character speaks only in hiragana.
Although there’s no digital release, you can read several chapters online on the Comic Walker website.
If you’ve never tried reading native material before, you probably won’t make it very far at first.
Joining the Absolute Beginner Book Club can drastically boost reading. You may want to push a little further along in grammar, but it sounds like you’re at a point where you should be reading native material every day (even if just a little) to give yourself that exposure to vocabulary (and grammar) in context.
You might be surprised how often native material will put random words in kana-only.