The coast is the edge of the land - so if you looked at a map, you'd say 'The coast is very straight here'. You can also say 'I'm going to the coast', just meaning that you're driving till you get to the sea. There are 'coastal paths' where you can walk along the edge of the land. It's more when you're talking about maps or geography.
A beach is specifically the bit where the sea washes - usually sandy(ماسه ای), or pebbles(سنگریزه). So that's the bit where you sit and have picnics, etc. Where the grass or something starts, it's not beach any more. So if something is washed up by the sea and can't get back into the sea, like a whale or something, we say it's 'beached'(به گل نشسته).
The shore is like beach, but beach is always a bit of sand or pebbles that is washed by the sea, whereas 'shore' is more general. So if a building is on the grass above a beach, you could still say it's on the sea shore.
So if you said 'I love the sea shore' you would probably mean the sea breeze(نسیم), the grasses, the birds, the whole atmosphere. If you said 'I love the beach' you'd mean actually hanging out on that sand.