Hi Caitlin -
The schedule is doable but you are only going to get bite sized chunks of each area. Here’s a bit of a suggested schedule - there are others of course, but this gives you what I think you want without spending too much time in your hire car!
It will take you two hours plus to get to York by train, rather longer if you drive.
York deserves more than a day or part of a day, but assuming you head on from there into the Dales, take the A59 through Harrogate and connect to the A65 at Skipton and head towards the Lakes. if you want to stop over you could stay in the little village of Clapham (very much not the London one!), or further on, Ingleton, which is a bit bigger. Both places have good routes up the flat topped hill that overlooks them, Ingleborough, if that’s your thing. (You’ll need the weather to play ball but in July you have half a chance and it will still be light well into the evening).
Moving on, continue on the A65 until just before Kirkby Lonsdale, turning right up Barbondale to reach Sedbergh via the A683. Here you could climb one of the Howgill Fells - Winder, Arant Hawe, The Calf, Cautley Spout, dependent on how much you want to do.
From Sedbergh take the A684 towards Kendal joining the M6 and heading north. Turn off at Junction 40 near Penrith and follow the A66 to Keswick, passing under Blencathra (some great if challenging hiking up that, it’s a real favourite of mine) and Skiddaw (big, but easier walking). If hiking Helvellyn, this is better done from Glenridding at the foot of Ullswater which will mean going back down the A66 if staying in Keswick and turning right on the A5091 to Glenridding via Thorneythwaite and Dockray. ‘Helvellyn By The Edges’ taking in the popular and crowded Striding Edge and the less popular Swirral Edge is a full day expedition and you’ll need a head for heights and be comfortable with a bit of exposure - look up some pics/videos on the net if you need to check the routes out.
The ascent of Catbells from Keswick is best done via launch from the lake (Derwentwater) to Hawse End and then climbing from there (a la Rick Steves) - Catbells, while ‘easy’ is not as benign as is often made out and you’ll need to get your hands out of your pockets near the top. Very do-able though!. Rather than catch the ferry back we retrace our steps back to Hawse End then walk back via Nichol End and Portinscale. Tea and cakes at Nichol End Marina and before that a new cafe near the house that was once a holiday home (forget the name for the minute but it’s visible through the trees and there are signs!) for Beatrix Potter (Mrs Heelis in these parts!).
Hope this gives you some food for thought. I also hope the weather co-operates and you have a great trip. Feel free to contact me direct if you need further advice or want to check up on other routes/options!