No, definitely don't "dress up" for the job.
I mean, wear NICE clothes, but wear appropriate clothes, not business clothes. Around here "dressing up" would be a polo shirt, breeches (as it's also a riding position) and comfortable walking/riding shoes. Be clean and neat.
Nice jeans, clean cowgirl boots and a "crisp" plain t-shirt is fine, or western shirt/blouse. Just look "nice" but still barn appropriate, don't overdo it. This is one job you CAN overdress for.
Now as you say "intern". I'm taking it this is a small private farm? If it's a larger facility I'd be more likely to dress up. I'm not used to a "barn intern" lol.
No make up (unless you usually do, I am always skeptical of people wearing make up in the barn lol) and a ponytail is fine.
You want to look professional and competent. And remember, professional is for THIS profession.