You must see the Castle District/Cathedral, then walk down the hill, cross the river and see asmuch of the Old Town as you can. Especially Old Town Square and the Jewish section (incredibly moving). Start as early as things are open and move fast - since Prague really deserves 3 days at a minimum.

I agree with NYT's itinerary. Make your way to the top of the Castle/Cathedral area and buy a ticket for entry soon after they open. You can easily spend a few hrs in the Castle/Cathedral area - be sure to go up the tower for great views.

Work your way down the hill after you exit by the old/small shops, browse vendors on the way down then cross the Charles Bridge to old town.

I've done several 1 day trips to Prague, and walking UP the hill after a day trekking around town is a pain. It's not a hard walk, but it is uphill.

If you want good photos of the bridge morning or late pm light makes for more interesting shadows.

You can walk, taxi, or take bus to the Castle/Cathedral area.

Kind of depends on what you like. The others have stated the obvious -castle with a walk over the bridge, walk to Old Town Square/Josefov.

Plan #2 - you could kill an entire day visiting churches, galleries, museums (skip the National, unless there is a special exhibit), and taking in the architecture, or, if you are musical, plan a visit to the Smetana museum, Dvorak museum and a concert. Take a break in a cafe, when you get tired, or have a beer... (in fact, some people might just stay in a beer hall all day!)

If it's raining, you could ride the #22 tram around town, taking in the sights, without getting wet. If it's nice, you could climb Petrin Hill (or ride the funicular). Go up Petrin tower with your camera & see all of Prague from there.

When I'm in a new place, I decide how much energy I have that day, and either run around full-force, trying too see all the sights, or I just decide to chill & soak up the vibes of the city. If you are feeling tired, I wouldn't push yourself too hard. You can see great views of the city from towers, parks, trams, etc. You could walk the Charles Bridge to get a taste of Mala Strana & Old Town, or you could take a river cruise, and get another perspective. The city is so beautiful & filled with culture, no matter what you choose to do, you'll end up with a fun time.

may I suggest you walk the Rotale Route in Prague.
http://www.prague.net/the-royal-route
I would probably do it in reverse order starting at the castle and working my way back (take the no22 tram up past the castle to the stop Pohorelec) it is about 2 stops past the main castle stop ,takes you away from the main crowds and allows you a nice amble down the hill to the castle area.In this area is the Strahov monestry and the Loretto which are both worth a visit but only if you have time.
You could easily spend a whole day in the castle area but give yourself a couple of hours or so before moving down towards Charles Bridge ,you will want to walk back and forth on this a couple of times.Follow the route to the old tTwn square and see the astronomical clock ,you can even climb the Old Town hall tower (they have a lift for those unable to climb),your route finishes /starts around the Powder tower and if it is open it is worth a visit (limited opening times).Finish your day with a drink or a meal at the magnificent Municiple House (Obecni Dum)You can even do a tour of this fantastic building if the weather is wet.
http://www.obecnidum.cz/web/en/homepage