Practice. I know that sounds really flippant but when it comes to paleography, it really is all about putting in the time and effort to get better. By the nineteenth century, script handwriting was more or less standardized to what we're familiar with and still taught today. For this reason, I don't know of any paleography courses or guides that focus specifically on scripts and alphabets after about 1800. It is worth looking into genealogy research online though as these websites often do discuss paleography somewhere. That said, most paleography guides and classes are geared toward helping modern people read medieval and early modern scripts as these alphabets were noticeably different from modern scripts. This is, of course, not to downplay the difficulties you're having with your own sources. Unfortunately, a certain percentage of the population has always had terrible handwriting. Griffonage is a real problem.

So, my tips are to practice. Make sure you're in a room with good lighting. Look at as many documents as you can and look at them carefully. Try to pick out individual letters in words you can read so you can familiarize yourself with the idiosyncrasies of that particular author's handwriting. Once you train your eye to recognize his/her letter formations, picking them out in the illegible scrawls will become easier. As frustrating as it may be, don't throw in the towel in a huff. Take breaks if you need them (in fact, it's good to take breaks so you don't strain your eyes) and then come back to it and start again. It will be slow-going but the results are often worth the pain and frustration. For instance, one of the letters I read recently was from a young gentleman to his uncle suggesting that the uncle (who was paying for the writer's education and that of his brother) remove his brother from school because 'he is too stupid to benefit from the lessons and his tuition would be better invested in the winter clothes that I have requested of you in my last letter.'

When I first started reading Scots secretary hand, I was lucky if I could transcribe 500 words in three hours. Now I can breeze through entire documents with (relative) ease. Don't get discouraged. It really can be mastered.

I don't personally study materials from this period, but I do work with them constantly as an archivist, and I often do also help people in the archives read them. The good news is it does get easier! The bad news is it's mostly just experience. Expert graphology is only earned in the historical school of hard knocks, in honest toil with secretarial ledgers and faded walnut ink diaries. But a few tips!

1. You do learn particular people's handwritings relatively quickly. There's an assortment of dead people (at the archives) whose notorious handwriting I can read relatively easily at this point, simply because I've read it a lot.
2. Context clues are big, sentences should make sense, period. If you're looking at a sentence where you've made out all but one word like "He made me a big f____ing sandwich" start trying to think of words that fit, and if they could concievably match up to the squiggles before you. Obviously, it's fortifying.
3. People in your era probably wrote in some form (however bastardized) of Copperplate round-hand, so get yourself familiar with that. Heck, you might even learn to write in it, knowing the strokes of handwriting can help you see where someone got off the rails and what letter they were allegedly forming.
4. Don't try to stare a word down and force it to decipher itself, this never works. Glance at words. If it doesn't come to you immediately, move on. Come back to it after a full pass of the document. Glance again. Might pop out ala the sailboat in a magic eye book, might not. You can't force it.
5. Ask someone else to look at it. Words that look like guonboblies to you might be obviously quotations to someone else. Team-transcriptions are always good practice, and how a lot of the crowd sourcing things work.

I can't really talk about European paleography, even though I suspect it is very similar to Japanese paleography, but basically if I want to read ancient documents I can use sorts of "dictionaries" (I don't really know more specific words in English) that list the contemporaries kana and their older versions, both typed and handwritten.

There are courses at university that teach paleography - I've never had one; I gather it is mostly medieval stuff. My friends who have done those mostly do them in summer seminars.

I did take a papyrology seminar. I wasn't good at it, but ten weeks with an instructor correcting my readings definitely improved my reading a great deal. The process was basically looking at a photographed papyrus, me attempting not even to translate, just name letters, and my prof pointing out why I was wrong. It sounds brutal, but it really helped. I don't know if you have access to such a teacher.