Suggestions for Western National Parks for an East Coaster

I am a senior in college outside of Pittsburgh and looking to take a trip over my spring break (March 16-20th). I have been to Tempe, AZ once but never farther west than Nashville besides that. Hiking/trail running has become a great hobby of mine recently and I am looking for some good national parks to do so. I loved the desert while in AZ but also love the idea of mountains and snow.

What are some of your suggestions for national parks that are:

1. Relatively inexpensive (major city close by to fly into/ lodging)

2. A good experience for someone that hasn’t been to a park

3. Somewhat safe for a solo hike/exploring

4. Would be good to visit in March

I have been looking at some of the UT parks, along with Olympic so far.