So I've been playing guitar for a little while now, and I'm self taught, hence the reason for this question. Or maybe its just a problem with everyone:

For certain songs that I want to learn, and there are a ton of these, there will be some difficulty in tuning the guitar. I'm not talking about dropped D. but rather tunings, in which you really have to tune up the G, B, and high e strings. i've done it on some occassions, but find it super sketchy because my strings will break when tuning, or playing, or sometimes, coming back down.

For example:
Don Ross's Berkley Springs is FACFCF. I'm reluctant to tune the B and high e strings that whole step up. I tried it the other day, and snapped a string.
Damien Rice's Dogs is DADADE. Tuning the G string up a step, and the B string up two steps is sketchy.

What can I do to rid myself of this problem? These are just two of many songs that i want to play, but can't because i dont want to spend an arm and a leg on strings.
use lighter strings, but then you gonna struggle when doing drop tunings, so there's a trade off. hope that helps.
Instead of tuning up, tune down and capo. So with your damien rice example, tune the "g string" down one step, and the high e, low e, d, and a strings down 2 steps, and then put the capo on the 2nd fret.

Partscaster/Tele into a bunch of pedals, a Maz 18 head, and a Z Best cab.
^ agreed, thats what I do. But I never usually use those tunings often, as I use light gauge strings. Lower tunings make the stringle rattle.
I remember when I first started guitar I tuned my guitar to what I thought was E but turned out being an A. I hit a C chord and broke two strings at once.
Next time you string your guitar, leave a little more slack on the strings. They'll feel looser in standard, but will let you tune up without breaking strings.

Alternatively, follow what the first two guys said and tune down then use a capo.