"I am never wearing lycra spandex shorts!" That sentiment has been heard the world over.
As for the bottoms - fortunately, lycra is not necessary. Sometimes it is helpful and sometimes it makes the ride more enjoyable. That said, traveling around the city, pretty much any clothing will do.
In fact, lycra is rarely seen on those that are getting about in the city. Now if the travel is taking you from town to town or is an all-day riding adventure, those bike shorts could make your day a lot more enjoyable.
And for the shirts, layers. Always layer unless it is very warm outside. This time of year is tricky. In the morning you could need three layers and a jacket. By the afternoon, maybe a couple layers will be all you need. Just be sure the first layer has fabric that wicks.
If you are braving the cold, a wool sweater will keep you warm and won't hold moisture. Wool sweaters from any thrift store are perfect. If it's windy, add a windbreaker. Again, all of these items can be purchased at a thrift store.
When your choose socks...skip the cotton. Truly, socks matter. Once you start wearing non-cotton socks, you will never go back to cotton. They are cute and fun and colorful! Unless you prefer the plain ones, there are those too. Check out the socks at BGI Indy or Bike Line.
The most important thing you wear is a helmet. Just do not ride without one. Period. Accidents can happen. Protect your head and that of your children. Get one that is certified. They aren't expensive and are certainly worth the investment. Read here for detailed information about helmets. *information courtesy of BGI Indy.