Car won't start.

Went to start my 2.4 L Sonata the other night. The
engine began to turn over and after about 3-4
seconds it stopped turning over. After that I tried it
again and again and again and it wouldn't even
begin to turn over. The battery seemed to have
power because the taillights, horn ,fan motor ,
radio all had enough power to function. We even
walk to a store and bought another battery for the
key fob. Still nothing. Eventually had it towed
home. Since then I had the battery checked and
they said it had a dead cell so I replaced it. Still
won't turn over. Also tried starting it with my spare
key fob. That didn't work. Even tried putting the fob
in the slot in the lower arm rest compartment and
still nothing. Also bought a new brake light switch
but if that diesnt work how can I find out  what is
wrong with it without having it towed Again to a
repair shop.