Why is my car taking a while to turn over?

are you sure they didn't say the starter was bad? unless your alternator went bad and drained your battery and you didn't re-charge the battery on a battery charger.when the alternator goes bad,all kinds of lights come on,BATTERY being the main one,then things quit working,like the A/C,steering,lights dim and so on and it drains your battery down and will finally strand you somewhere.if none of that happened,the alternator has nothing to do with your engine turning over unless your battery is weak or bad.

Sounds like your alternator could be going out, or could simply be a older battery either way you have a charging system issue with the hard to start. The blinker randomly blinking fast is most likely due to a bad connection in the bulb socket.