You'll be fine going with stock if that's all you do and you aren't planning on doing any overclocking. Now you could always go with a cheap aftermarket cooler like the Cooler Master Hyper 212+ just for some extra cooling to throw on if you have the spare cash.
But otherwise if you want to save as much as possible then yes the stock heatsink is fine for what you want to do.
For future upgrading I'd strongly advise you to go 2500K. You don't have to overclock now, but you'll see what I mean in three years or so. When the 2400 would have started feeling balky, you'll just push the 2500K a few hundred mhz and have a whole new CPU for no extra cash. If you live near a Microcenter, it's the same price as the 2400!
A stock sink'll be fine for now, with a non-OCed CPU, but you may want to get one for future OCing of the variety mentioned above (life-prolonging instead of performance-pushing). It can be quite annoying to install a heatsink on an already-assembled build: you have to take out the mobo, with all the fiddling and unplugging that implies, remove the old thermal compound, apply the new compound and sink, and then wrangle the whole thing back together.
Get a sink now, and get a 2500K to go with it, preferably at Microcenter - you'll save $40. The 212+ is good. This will save you money in the long run by obviating an upgrade or two.