I am going to spend about 10 days in PHX but I am having difficulty visualizing the different parts of the urban area and their relationships with known points of interest.
I am an avid hiker and would like to base myself in a part of town close to hiking trails. Which part of town do you recommend? I shall hire a car so proximity is somewhat relative.
Thank you for the tips.

I would go north or east. North gets you the Phoenix Sonoran Preserve (near I-17 and Dove Valley), Deem Hills, and Spur Cross. Look for lodging in Anthem, Cave Creek, Carefree, or far north Scottsdale.

Note that the area around and south of the I-17 and Loop 101 junction is NOT a safe area, but just north around I-17 and Pinnacle Peak or Happy Valley is fine and has several lodging options. Also, the areas around the Loop 101/AZ51 interchange and the northwest "corner" of Loop 101 (where it goes from an east/west to a north/south freeway in Peoria) are fine too.

North Phoenix also gives you an easy drive to Sedona, Prescott, and Flagstaff, which all have great hiking in the mountains.

East has the Superstition Mountains, McDowell Mountains, Four Peaks, and Salt River area. Look for hotels in Mesa, Fountain Hills, or Scottsdale. Chandler and Gilbert may also be viable, but are a little further away.

This also gives you good access via AZ87 (locally known as the Beeline Highway) to Payson and the Mogollon Rim.

Piestewa Peak (near AZ51 and Glendale Av) and Camelback Mountain (near Tatum and Lincoln) are pretty centrally located about 5 miles north of the airport. These are the two major urban mountains that everyone thinks about when they think hiking in Phoenix.

When will you be here?

edit: Here's a map with some of the more popular hiking areas circled:

Lost's map is an excellent guide for your question.

With a car, however, just stay in a nice hotel in Scottsdale. The road system is very good in the valley, and the only places circled above that are a longish drive from someplace in Scottsdale would be the White Tanks out in the West Valley.

Plus if you pick a hotel on the Camelback Greenway, you can do some lovely biking or flat hiking along the length of Scottsdale, along the canals, through the Scottsdale lakes, etc.

For example, you can choose a hotel such as the Hyatt Gainey Ranch, and use their bikes to ride down the Greenway to Camelback Mountain, hike to the summit and down, eat lunch in Old Town, and ride your bike back to the hotel. The ride will take you past the lakes, through golf courses, along a canal, and through Old Town.

Thank you so much guys for the detailed input.
LOST: I will be there in early April. One last question: would it be feasible and safe to hike to and from Camelback mountain / Piestewa Peak from my hotel?

I would say hiking the mountain is ok as long as you do it early (meaning start right at sunrise. Both mountains take about two hours to hike round trip if you're in decent shape). April is a transition month where high temps can vary between 80F-100F and overnight lows between 60-70. As far as personal safety both Piestewa and Camelback are in safe areas and I would not have an issue walking those neighborhoods at any time of day or night.

If you're coming from a cooler or more humid climate beware that you'll need 2x-3x more water (start hydrating before you arrive). And sunscreen. Lots of sunscreen.

I think the Phoenix Sonoran Preserve, McDowell Mountains, and Camelback (from Echo Canyon TH) / Piestewa Peaks would be the best areas to get a taste of the desert and aren't too far away.

There are some amazing hotels in walking distance from Camelback Mountain. The Royal Palms, The Phoenician, Scottsdale Camelback Resort are all right there.

I think it really depends on budget. As noted, there are places to hike almost everywhere. In addition to all of the mountains listed, I wouldn't ignore The Maricopa Trail, which is all over the entire area, even in metro areas. Parts of it are alongside canals. No, it may not be as scenic or well-known as Camelback (I have hiked that and agree is is nice), you'll see an entirely different side of the area. At times you'd be able to look right into some backyards. You can get a sneak peak into the Biltmore Resort from it.

Sure, if you have $300+ to spend per night, then definitely look into the Phoenician or similar. But if you want to put that into a third, then I suspect you'll be looking more toward Tempe, or other suburbs for more reasonable hotel rates. You might be driving, but so long as you avoid rush hour, it usually isn't too bad.

The Maricopa Trail doesn't go near the Biltmore.

The Maricopa Trail is a 300 mile loop around the Phoenix area which I believe is about 80-90% complete. While I'd like to eventually hike the entire trail piece by piece, I don't know that hiking a small segment of it would be worthwhile for a visitor, unless they're wanting to use it to connect between two of the regional parks.

There's many other trails that would offer more bang for the buck for the OP.

This. I would pick a hotel based on it 1) having the amenities and quality I desire and 2) being priced reasonably for my budget. It being within walking distance of a particular trail would be at best a third priority. I'd plan on driving. Outside of rush hour traffic is not prohibitive. Rental cars are often not too expensive at PHX -- and when they are Lyft/Uber works well.