How are you getting to Belgium? If by plane, and I assume an arrival at Brussels airport, then by train from the airport to Liege is about 50 minutes. Take a train to Leuven, roughly 20 or less minutes, and then a connecting train to Liege, about thirty minutes.
If you show up early in the week, you can also watch La Fleche Wallone. I'd recommend trying to get a place on the Huy finish line to watch the riders slog it out on the hill to the finish.
As for Liege-Bastogne-Liege, the meaty climbs begin after Trois-Pont. Problem is that Trois-Pont is an hour away from Liege by train and it seems unlikely that a selection would be made a 100 kilometres away from the finish, unless of course someone does their best interpretation of Bernard Hinault circa 1980. Though if you are bringing your bike and would like to do some cycling, the Trois-Pont area is right in the Ardennes and scenically quite beautiful.
If you want to guess where the action might happen, maybe the 9th climb, Côte Roche auf Faucons or the 8th climb Côte de la Redoute would be decent bets to see where a selection might be made. Actually, the race crosses near the 9th climb, so you could actually catch the race twice, once when the cyclists make their way to Bastogne and the other time as it draws to a conclusion on its way to Ans. It would be a about a four hour wait if you do this though. Failing that, Ans, the finish line, is close to Liege. I honestly don't know if Liege-Bastogne-Liege can offer up the same atmosphere as either Flanders or Roubaix, though I say that from not having experienced any of them first hand.
if you will be travelling by train, the website for belgium's rail line is nmbs.be. It has a user friendly english option.