Riana,
Since you say most have not seen Yosemite my suggestion is this:
On the first day get a permit from Glacier Point for a Little Yosemite Valley
PASSTHRU. The backpack trip you can take the Panarama Trail to Nevada
Falls and then to LYV. From there continue up the JMT towards Half Dome
and turn right away from HD. Stay at the Sunrise Creek/Clouds Rest Tr. Jct.
This is 9 Miles with total of 2000ft gain. On second day you can get up
early and concur Half Dome. From there you have options.
a) You can stay there and go for Clouds Rest
b) You can stay there and nappy
c) You can continue on the JMT and head towards Echo Valley/Merced Lake
If you choose option C it is a total of 11 miles if you do not go to Merced
Lake back to LYV. So on Day 2 you could get to granite areas looking
in Merced River Canyon or you could ...
Anway, the Merced River Canyon around Lost Valley and Moraine Dome
is spectacular.
You can finish your trip by hiking down to Yosemite Valley.
(to start the trip you could shuttle yourself since you have day 1 to
explore the valley and check out Glacier Pt.)
Or you can hike back up to your vehicle at GP.
Anyway, that is my suggestion simply for the fact that you said not
many have been to Yosemite and you want to do Half Dome.
(and I think the Merced River Canyon is something to behold)
Please report back as to what you do. (and if you choose this get to the
permit station EARLY the day BEFORE you start your trip)
If going to Vernon then... as said ... permits should not be an issue.
Have fun.