I just started my first semester of beginning piano. I seem to have trouble counting in 3/4 time. I know it will come with practice, but does anyone have any tips for practicing non-standard time?
I wouldn't consider 3/4 to be "non standard".
My advice would be to find recordings of pieces in 3/4 and count along to them.
The "oom-pah-pah" pattern is a very common way to arrange a 3/4 backing. I suppose the most direct example is "Oom-pah-pah" from the soundtrack to the musical "Oliver!" - so listen to that first. In the chorus, they actually sing "Oom-pah-pah". "Oom" is beat one; "pah-pah" is beats two and three. In the verses you can practice without that crutch.
Start by spotting the downbeats, the "oom". Most of the time they are pretty obvious - a bass instrument or a drum will be marking them. Call out the downbeats by saying "one".
Once you can do that, fill in beats two and three. Often these will be emphasised by a higher pitched instrument like violins or the higher strings of a guitar. Call out these by saying "two three". And you've got it!
There are plenty more examples in pop/rock. Try "Norwegian Wood" or "It's A Man's Man's Man's World", or any one of thousands of country ballads. In classical music, of course there are lots of waltzes, minuets, etc.
Of course, not every piece in 3/4 uses an "oom-pah-pah" arrangement but once you can count 3/4 against an oom-pah-pah backing, you should be comfortable enough with a three-beat bar to apply it to other pieces.
Once you can reliably count time on recorded music, it should be quite straightforward applying that to music you are playing yourself.
It may also help to use an electronic metronome -- the kind where you can set the time signature to get a different click on the downbeat. You can buy them as dedicated hardware, or use phone/tablet/computer apps.