OK, since everybody is doing this...
Alive In Athens is the point where you should start. It's the point where I started, it's the point where everyone I know started and it's the point where I'd let your grandmother start. It's the quintessential live album, it's got the majority of the bands' songs to that point on it, and it's also extremely affordable for a three disc set.
Once you have it, you may ask yourself if there's even a point in buying the older studio material. I say do it, it's worth it. My personal favourite album of the band is Dark Saga, but maybe you'd want to get Burnt Offerings first, because only very little of it made it to Alive In Athens. Jon Schaffer said he personally hates the album, but every fan I know of disagrees with him. It is a very aggressive, angry album, but musically very brilliant.
Of course, it's alway a matter of preference. If you enjoy thrash metal, then you should get Night Of The Stormrider, which is considered by many to be a classic album of the genre. Their debut, Iced Earth is also not to be tampered with. It may still be a bit unpolished, but that's exactly what makes it worth having.
If you prefer power metal à la Blind Guardian, then you will want to get hold of Horror Show. It is not as popular as the previous albums, mainly because the thrash edge the band always maintained is mostly gone in favour of an epic/bombastic one. It may have a couple of forgettable tracks on it, but generally, it's a very good album and don't shy away from it.
The power metal trip is continued on The Glorious Burden, which is the most controversial release of the band. If you've grown used to Barlow's vocals by this point, you may not want to familiarise yourself with a new -and very different- singer, especially if he's already been booted in favour of Barlow again. Frankly, if you need to skip one Iced Earth album, let it be this one. It's got a couple of pearls on it, but all too often, the songwriting is very weak, and Owens, as good as he is, can't always make up for that.
Which leads us to the present day, with Framing Armageddon, which is very much like Horror Show, very unlike Something Wicked This Way Comes, and fortunately, only like the good bits of Glorious Burden. The songwriting has improved dramatically from its predecessor, but if you're not into that epic/multi-layered-chorus-bombast thing, you might hate it.
Oh, speaking of Something Wicked This Way Comes. It's a great album, in my opinion, and I very much enjoy the songs Forostar called "fillers" too, but you may indeed ask yourself if it is worth it to get this album if ten songs are already featured on Alive In Athens. My say is... yes, it is.
If you don't own any albums or only one or two yet, that new box set in its entirety is a good bargain. It looks very neat and the price is justified. I won't get hold of it, though, because I already own all the albums, and I don't think the "bonus material" is worth it at all.