My friend is starting D&D and his first character is going to be a swordmage. How many spells do they get at level one and how many do they get per level up?
The answer is 0. Sort of... But not really. Ok, they're called spells, but they don't act like spells in previous editions, calling them spells doesn't make them spells.
Characters in 4e don't exactly get spells. You get powers. Sometimes they're called spells, sometimes they're called exploits, but that doesn't really change the fact that they're, ultimately, powers. L1 characters generally get 2 at-will powers, 1 encounter and 1 daily power. You also get class features.
The full Swordmage is detailed in the Forgotten Realms Players guide, and the text can be found in the DDI online compendium and is accessible with a subscription to Dungeons and Dragons Insider.
They get the same amount of powers as any other class. (They're called Spells flavourfully for Arcane classes, like how Martial classes have Exploits, but they're all just powers.)
For the first level, see this question. For other levels, refer to your level-up table on page 27 of the Player's Handbook: this contains what you gain at each level, plus what you have total at each level. We probably can't legally reproduce this table for you.