For nature landscapes while traveling, hiking & family/friends parties

Go for the T3i only if you have enough cash left over for some extra lenses.  Moving from the T3 to T3i is not much of an upgrade (Accord DX to Accord EX), so go with what you can afford.  The camera body is not going to make that much of a difference how your photos look.  What will be important is lens selection.  Most beginners will get the kit with the cheap 18-55mm and think nothing of it.  For some really impressive landscapes you may want something wider like a Canon 10-22mm or Tokina 11-16mm.  These are very good, but have limited "real life" usability.  Another idea is the Canon 15-85mm.  Excellent all around with a nice range.  Only draw back is that this lens, like the kit lens, doesn't let in much light when zoomed in.  You will have to use a flash when shooting indoors or crank up the ISO.