About the weeds, generally speaking yes, pulling them out should do the trick. However a picture or two of the weeds and yard would really help with any further diagnostics, as would any information related to your USDA grow zone. Some weeds are more vigorous growers and can survive underground through gargantuan root systems that really put up a fight.
As for buying tools/supplies, I have had great success using a weed whacker or its beefier counterpart, a bush whacker if things get too hairy. The auger looks like a good deal but from what I have heard its utility is limited to more of a specialized digging tool.
About the compost, depending on what you have in it and in what amounts the process can take a few weeks to many months. I would recommend adding more soil and fallen leaves into the pile to increase the microorganism count in your bin. You should also be stirring the pile at least once a week, adding water whenever it is dry or whenever you add a large amount of matter to the mixture. I would also highly recommend adding as many earth worms as possible, they work wonders. Aside from decomposing the compost faster they also leave deposits of droppings that plants of all kinds benefit greatly from called "worm castings".
I am not super knowledgeable about lawns as I am more of a tropical/ indoor plant person however
AngryGerbil
seems to have this lawn thing down if I remember correctly. I do know that the best time to fertilize is during the active growing season, for me that is pretty much over due to my latitude however depending on where you are on the USDA agricultural chart you may be able to squeeze this project in before fall. If not I would just keep enriching your pile during the year until next spring when it is the optimal time for plant development.