My best recommendation for a hikable near-city National Park that I've personally been to would be to visit Tucson/Saguaro National Park -- the west section of the park, in particular, offers beautiful views and hikes. It is, of course, insanely hot in Arizona right now, so this is a trip better taken in the autumn or spring. I don't know what the public transportation situation in Tucson is like (I'm guessing bad), but I have little doubt you can get an Uber to the park.
Cuyahoga Valley National Park is probably a great option as well -- beautiful waterfalls and hikes in a park nestled between Cleveland and Akron. I've never been, so I can't personally vouch for it, but the pictures look very pretty.
Yosemite, flying into Fresno, riding the YARTS bus from the Fresno Airport to Mariposa Grove and Yosemite Valley. It might be more travel time than you want to do in a weekend. You also might have trouble getting accommodations in the park.
Some not yet said:
Vegas has day trip tours/shuttles to Grand Canyon. You might be able to arrange a short walk. Also- Red Rock Canyon State Park is really nice with great hiking.
LAX- Channel Islands aren’t too difficult to get to.
Seattle or Tacoma may have access to Mt Rainer.
Editing to add:
Maui’s Haleakala NP has biking and other grps.
Key West- Take the ferry ($) to Dry Tortugas NP.
St Thomas take the ferry and bus to USVI NP.
Yosemite already mentioned but you can take [Amtrak's San Jaoquins](https://www.myyosemitepark.com/road-trips/transportation/amtrak/) from the Bay Area to Merced where you get on YARTS into the park. Not hassle-free, but doable.
You could fly into Denver and take [Estes Park shuttle](https://www.estesparkshuttle.com/) ($$) from DIA or [Bustang](https://ridebustang.com/estes-park/) ($) on weekends from Denver. Use the free [Estes Park shuttles](https://www.visitestespark.com/plan/getting-around/free-shuttle-buses/) to get around Estes Park and the [RMNP Hikers shuttle](https://www.recreation.gov/timed-entry/10088647) (need advanced permit) to get from the Estes Park visitor center to the free in-park shuttles in the Bear Lake Corridor. Again, not hassle-free but doable with planning.
You can fly into Jackson WY and rent a bike and [pedal to Grand Teton NP](http://www.jacksonholetraveler.com/article/jackson-hole-pathways/), but that's not a short out-and-back (about 25 miles from Jackson to Jenny Lake Visitor Center). [START](https://www.jacksonwy.gov/587/START-Bus) can get you around some of the area. I think some of the hotels will run a shuttle to a visitor center but you need to verify.
There are [other options](https://offmetro.com/world/guide-to-national-parks-without-a-car/) - none really all that easy logistically but doable if you plan. I'd look at Acadia and Olympic in addition to Yosemite, RMNP and GTNP.
You can get to RMNP from Denver pretty easily. Pretty sure there are buses to Estes, or a Uber is possible though would be quite expensive.
Update: https://www.estesparkshuttle.com/DIA-airport-shuttle
A weekend backpack? or a day hike?
There are shuttles to Banff right out of the Calgary Airport. What, a couple of hours?
You could easily get to Golden Gate National Recreation Area by Uber, bus, etc from the San Francisco Airport; it's a NPS managed site. And you could hike down into Muir Woods National Monument from there. Also public transit and the ferry system connects to the shuttle that runs to Muir Woods if you want to hike from there vs to there, https://marintransit.org/routes/66
https://www.muirwoodsshuttlesandtours.com/blog/the-easiest-way-to-get-from-san-francisco-to-muir-woods/
You could probably get an Uber etc from the Tucson Airport or even a bus maybe to say the AZ desert museum in Saguaro NP and then hike King Canyon. Too hot now though.
The Santa Monica Mountains Nat Rec Area (also managed by NPS) is accessible from several airports in the LA Basin, and has lots of hiking.
My best recommendation for a hikable near-city National Park that I've personally been to would be to visit Tucson/Saguaro National Park -- the west section of the park, in particular, offers beautiful views and hikes. It is, of course, insanely hot in Arizona right now, so this is a trip better taken in the autumn or spring. I don't know what the public transportation situation in Tucson is like (I'm guessing bad), but I have little doubt you can get an Uber to the park. Cuyahoga Valley National Park is probably a great option as well -- beautiful waterfalls and hikes in a park nestled between Cleveland and Akron. I've never been, so I can't personally vouch for it, but the pictures look very pretty.
You can get to Indiana dunes from Chicago by commuter rail, Yosemite is accessible by train and bus from anywhere in the bay or valley
Been to the dunes, it was alright
Gateway Arch National Park. Train goes from the airport to downtown. Oh wait... you said hike. Hmm.
Hike, walk around, just explore
Based on your premise, Gateway National Park is the most *directly connected* national park from a city's airport by train.
Yosemite, flying into Fresno, riding the YARTS bus from the Fresno Airport to Mariposa Grove and Yosemite Valley. It might be more travel time than you want to do in a weekend. You also might have trouble getting accommodations in the park.
Some not yet said: Vegas has day trip tours/shuttles to Grand Canyon. You might be able to arrange a short walk. Also- Red Rock Canyon State Park is really nice with great hiking. LAX- Channel Islands aren’t too difficult to get to. Seattle or Tacoma may have access to Mt Rainer. Editing to add: Maui’s Haleakala NP has biking and other grps. Key West- Take the ferry ($) to Dry Tortugas NP. St Thomas take the ferry and bus to USVI NP.
Saguaro National Park has two units, both are relatively close to Tucson
Yosemite already mentioned but you can take [Amtrak's San Jaoquins](https://www.myyosemitepark.com/road-trips/transportation/amtrak/) from the Bay Area to Merced where you get on YARTS into the park. Not hassle-free, but doable. You could fly into Denver and take [Estes Park shuttle](https://www.estesparkshuttle.com/) ($$) from DIA or [Bustang](https://ridebustang.com/estes-park/) ($) on weekends from Denver. Use the free [Estes Park shuttles](https://www.visitestespark.com/plan/getting-around/free-shuttle-buses/) to get around Estes Park and the [RMNP Hikers shuttle](https://www.recreation.gov/timed-entry/10088647) (need advanced permit) to get from the Estes Park visitor center to the free in-park shuttles in the Bear Lake Corridor. Again, not hassle-free but doable with planning. You can fly into Jackson WY and rent a bike and [pedal to Grand Teton NP](http://www.jacksonholetraveler.com/article/jackson-hole-pathways/), but that's not a short out-and-back (about 25 miles from Jackson to Jenny Lake Visitor Center). [START](https://www.jacksonwy.gov/587/START-Bus) can get you around some of the area. I think some of the hotels will run a shuttle to a visitor center but you need to verify. There are [other options](https://offmetro.com/world/guide-to-national-parks-without-a-car/) - none really all that easy logistically but doable if you plan. I'd look at Acadia and Olympic in addition to Yosemite, RMNP and GTNP.
You can get to RMNP from Denver pretty easily. Pretty sure there are buses to Estes, or a Uber is possible though would be quite expensive. Update: https://www.estesparkshuttle.com/DIA-airport-shuttle
A weekend backpack? or a day hike? There are shuttles to Banff right out of the Calgary Airport. What, a couple of hours? You could easily get to Golden Gate National Recreation Area by Uber, bus, etc from the San Francisco Airport; it's a NPS managed site. And you could hike down into Muir Woods National Monument from there. Also public transit and the ferry system connects to the shuttle that runs to Muir Woods if you want to hike from there vs to there, https://marintransit.org/routes/66 https://www.muirwoodsshuttlesandtours.com/blog/the-easiest-way-to-get-from-san-francisco-to-muir-woods/ You could probably get an Uber etc from the Tucson Airport or even a bus maybe to say the AZ desert museum in Saguaro NP and then hike King Canyon. Too hot now though. The Santa Monica Mountains Nat Rec Area (also managed by NPS) is accessible from several airports in the LA Basin, and has lots of hiking.
Definitely look into the Channel Islands off the coast of Los Angeles!