Zion is definitely one of my favourite places... I think I've visited there... hmm, let me see... probably close to 10 times now.
Definitely Angel's Landing is a must, so long as you are ok with heights (besides, there's a chain to hold on to for the exposed sections). Not very long, perhaps 3-4 hours at most. Plus it is practically paved most of the way. I think I've probably hiked this trail close to ten times now.
Observation Point is also excellent - in some ways, better scenery than Angel's Landing, but as another poster said, not as "exciting" (i.e. exposed). Mixes main canyon views with some neat side canyons. The point itself (i.e. Observation Point) provides a great way to see Angel's Landing from above (the top elevation of the Observation point hike is a fair bit higher than that of Angel's Landing).
A nice little intro hike (just an hour or two) is a short trail near the main campgrounds - I think it is called the watchman trail. Goes up to a bench below a prominent point called the Watchman, and looks down over Springdale (the little town outside the park).
Other very nice hikes:
-the hike to the Emerald pools is nice, and not that long.
-if you continue up (past Angel's Landing) on the west rim trail, you can access the high country. In april there may well be snow up there. If you want a long dayhike this'll fit the bill. Excellent scenery, of course.
-If you like off-trail adventures and a bit of ropework, there's left fork north creek. Goes through an area called 'the subway'. You should do a little bit of reading on canyoneering before you tackle something like this, though.
I have a couple of good hike writeups for Zion on my webpage (one of which was when I took my Dad on a desert southwest trip). Here's the links:
A bit of writeup on taking my dad to Zion, plus the hike to the Emerald Pools...
Hike with my dad up Emerald Pools Trail (2003)
Another page from the same trip, this time describing the hike to Observation Point:
Hike up to Observation Point (2003)
Here's a page with a good description of the hike up Angel's Landing. It is taken from a page in a trip we took in the spring of 2005 (mostly a grand canyon backpack trip, but we did Angel's Landing on the last day):
Angel's Landing Hike 2005
And one more description of one of our Angel's Landing hike, just so I've got all the angles covered:
Angel's Landing Hike 2004
Hope you find all of this useful!