The main trails at the canyon are very well marked and maintained (although still trails, of course) and very easy to follow. They are heavily used during the day and you are almost always in sight of other hikers. Doing short day hikes on the main trails is a great way to experience the canyon - alone or in a group. I have day-hiked there on my own many times.
You do need to hike smart and know your limits (with or without a group). As the poster above mentions, it is harder going up than it is going down. It is also much warmer at the bottom, than at the top, but you should not be trying to go to the bottom and back as a day hike anyways (alone or in a group). Generally good advice is to wear appropriate shoes/boots, carry plenty of water, wear a broad-brimmed hat, carry some snacks (salty ones in particular) and - if you are going in summer - avoid being on the trail during the peak of the day.
Here are my favorite short day hikes off the South Rim:
South Kaibab Trail to Cedar Point (1.4 miles one way) - Excellent sweeping views across the canyon and a good chance to see wildlife.
Hermit Trail to Dripping Springs - Goes into a major side canyon to a pretty little seep spring. A bit longer and steeper than some of the others.
Bright Angel Trail to 1.5 or 3 mile rest houses (1.5 or 3 miles one way) - Probably the most popular trail (lots of other people) to small stone ramada overlooks.
Have fun and enjoy your hike!