Leave the cores on the grass! Water deeply only when you are able to see your footprints left on your lawn/grass. Great indicator. Use a spade to see how deep the water reached into the soil profile after 15 minutes. It should be a minimum of 4". 6" is better depending on the porosity of the soil.
Allow the lawn to dry out and don't water again until you leave footprints on your grass! Watering a few minutes once every day is promoting shallow roots and the first time you forget to water might stress your grass to the point it will go dormant. Stressed grass can not compete with highly competitive weeds!
Another common mistake is to cut your lawn too short. 2 1/2" is too short for the grasses I am familiar with, here in the Pacific Northwest. I'll go look up your profile to see what types of grass with which you might be dealing. Raise your deck up as high as you can to see if you can cut your lawn no lower than 3". The soil stays shaded preserving moisture and weed seeds don't get enough light to germinate.
If you have the same types of lawn grasses (bluegrass, fescue...) these species genetically have huge root systems. If there is not at least 3" of photosynthetic top growth the plant can't make enough food to feed the roots and you have stressed, whimpy grass allowing weeds to get started in the sparse lawn.
Here is a tip that will make a humongous difference: Fertilize at least twice a year with a slow release organic fertilizer. I used Dr. Earth Lawn Fertilizer and was blown away with the difference in my lawns! A bit more expensive but it lasts longer, adds bacteria that aid thatch-breakdown, and beneficial fungi that help the plant uptake nutrients, water.
Don't mess with moss-killer, that will lower your pH and lawns love a bit higher pH (6.5-6.8). If you want a beautiful lawn you should definitely get your soil tested. Only add lime if the pH is below 6.5 and follow the instructions! Moss is an opportunist...if you water everyday, if you have any bare spots, shade with sparse grass, moss will grow! If you get your grass healthy, mow high and allow to dry out in between watering (about 1" per week...you'll have to train your grass in steps to get deep roots to last a whole week)! I only watered my 'trained' lawns, once per week and they were thick, cool, soft and no weeds. Aerate once per year minimum. You might want to up it to a second aeration for a few years. If you have more than 1" thatch you will need to do it and then reseed, mulch, roll...etc. Lot of work but really is for drastic measures as it ruins your lawn thus having to start over.
Hope this helps, there are more things one needs to learn in order to have a beautiful and healthy lawn! Stay away from herbicides, pesticides and quick fix instant fertilizers that call themselves 'plant food.' Plants make their own food only needing us to add certain nutrients because we start with poor soil, compacted during construction, poor mowing practices.
There are a bunch of great question/answers concerning lawns on this site that will give you even more knowledge and tools that will help you achieve mastery over this 'beast', the lawn.
