The start of April is fine. It’s the tail end of tourist season. The weather is generally pretty nice (lower to mid 70’s).
Others may disagree, but I honstely don’t think Phoenix (and surrounding suburbs) has much in the way of “must see” attractions.
But If I were pressed here are some suggestions:

Metro Phoenix:
Desert Botanical Garden
Rawhide
Heard Museum
South Mountain Park
Papago Park
The Horny Toad
The Satisfied Frog
Old Town Scottsdale (very touristy)
Biltmore Fashion Park (if you’re a [window] shopper) in Phoenix &
The Borgata of Scottsdale (if you really wanna see how the “other half” shops)

Outside of Metro Phoenix:
Arcosanti (1 hour north on I-17)
Casa Grande Ruins National Monument (1 hour drive south on I-10)
(City of) Sedona (2 hours north [mostly on] I-17) &
Oak Creek Canyon
Apache Trail Senic Byway (1 hour Southeast of Phoenix).

I traveled to Phoenix alone in November for work. On my free day, I drove to Sedona and did a Pink Jeep Tour. It was about $80 or so for a few hours (I thought it was a bit pricey), but since I was by myself I got to sit shotgun in the jeep and had a fantastic view. The rest of the group was really nice, too. I am from Pennsylvania and it was my first time “out west” so the drive in itself was a great time.

Here is a link to my trip on FLickr.
http://www.flickr.com/photos/pilarkitty/sets/1293901/

In April it will already be pretty hot, so don’t plan on wearing pants, but shorts.

As far as where to eat:

-Breakfast: Waffle House

-Lunch: Port of Subs, Chompie’s

-Dinner: LoneStar Steakhouse, Macaroni Grill, La Parilla Suiza

As far as where to visit:

-Phoenix Zoo
-Heard Museum
-City of Sedona
-Scottsdale Fashion Square
-Grand Canyon
-Glendale shops

Enjoy your stay!

Places to go:

- A Friday or Saturday ‘Night on the Town’ on Mill Ave. in Tempe (it’s the ASU hang-out area)

- Tour or catch a game at Chase Field, home of the Arizona Diamondbacks

- Fajitas Mexican Restuarant on the Black Canyon is a terrific restaurant

- Castles N’ Coasters, near Metrocenter, has four mini golf courses, a huge arcade and several rides including a looping roller coaster.

- Saturday April 1st is the 2006 Arizona Book Festival near the Arizona capital (my writers group will have a booth there)

- A drive for a day trip to Sedona is good. Lots of places to stop along the way too

- If you’re looking for events, AZCentral.com has an events calendar posted on their web site.

- ThingsToDo.com has….well, a lits of things to do by state

If you are looking for something that might be different from other things listed so far, try the Broadway Palm West, a dinner theater in Mesa. Good food and a good show – all musicals (I think Funny Girl will be running while you are there).

Other than that, the suggestions above. I loved the day trip we took to Sedona – wished it was longer!

I went in January. I didn’t make too many plans and was glad. In the end I just felt like being lazy at the resort I was staying at. To me, Phoenix is about hanging out and not doing much. That was never my travel style until I moved to Chicago, where it is non-stop. I love Chicago, but it was very good to slow down.

I did like Old Town Scottsdale, definitely touristy, but is one of the few places I knew of where you can walk around. If you like a scene, the James Hotel is a fun place.

If you like a college atomsphere, check out the ASU campus. In downtown Tempe you can shop and eat and drink with the college students. I’m not sure what sports go on in April, but it would be interesting to see some sort of game there. Football would be the best – Sun Devil stadium is cool, but alas, not for April.

I don’t golf, but I golfed anyway when I was down there. Just rented some clubs and hit the high-end golf course at the resort. I work for a hotel company and they gave me sweet deals on everything – I’m not this rich. The golf in Phoenix is expensive but if there is any way you can do it, to me it’s a must-see or do. Golf is big part of the tourism culture down there and the views from the course were stunning.

Definitely if you can do a little day trip do go to Sedona as others have also mentioned. It is lovely!!!