Obviously the string letters correspond to a space where a note goes in standard notation, but that's only for open notes. Is there a pattern to remember which frets represent which notes at any tuning, or do I have to learn them each individually?
You will have to learn each note's location for each separate tuning. That's all there is to it, unless you're simply tuning everything up or down by the same number of semitones.
Seems like you're a beginner, so go for standard tuning, and bear in mind that there are (eventually for you!) many different places to find exactly the same note on a guitar. It doesn't need to concern you for now.
I learned the positions of notes on the fretboard with four mutually supporting approaches:
Absolute position: I used flashcards to drill myself on the positions of each note on the fretboard. This enabled me to move up and down the strings.
Relative position in scale patterns: I learned which scales corresponded to which key signatures and then I mapped each note in the scale of the tune to the scale degree (the number of the note in the scale from 1 to 7). Then I could play tunes by moving from string to string in the scale pattern. In order for this to work you need to know where the root note is in the scale pattern.
Relative position in intervals: I learned the relationship of each interval between the strings on a given fret. For example, the open strings on the guitar are EADGBE which means that the intervals are in fourths up to the GB strings which are a third. I learned just the chord tone intervals of major and minor thirds, fifths and octaves and thanks to the scale pattern work, could fill in the rest of the notes.
Sight reading: Finally, I just started sight-reading. Painfully slowly until I got the tune right and then gradually increasing the speed. This is the hard part and takes the longest. If you know your scale patterns you will find that most tunes have a sweet spot on the guitar where you can grab all the notes with minimal shifting up and down the fretboard.
I recommend you finding a good instructor. Depending on where you live, i might suggest someone. The difficulty lies in the fact that you are learning two different aspects of music, the theory(intervals, harmony) and playing guitar, which is essentially an endeavour of practical nature(part of which is knowing the location of notes on the fretboard). Both aspects feed of each other and it might take a while(months) but you'll get it all! Its very hard figuring it all out on your own, and it might take too long. So look for a good teacher!
I enjoy studying music, so for me it isn't work. I read a lot, memorize and experiment, and research things I don't yet understand until I get enough information so that I can understand. This forum can help immensely. I've been studying for a few years now and I find that there's up-hill climbs and plateaus for me in the development process, sometimes I can comprehend and sometimes I can't quite. But that's the way my life goes in general so I just keep working on it till it makes some sense. That said, the first thing I'd say for you to do, would be to memorize the note locations on each string from the first fret to the twelfth fret, take it one string at a time and take a little time to get used to using this new information. And hopefully you'll enjoy the learning process and feel inspired to continue. Simple as that.