This is such a fun age... when they start getting around and exploring.
I would start by finding a "group" like MOMS Club, MOPS, your early childhood PTA (ours is for kids 0-5), or neighborhood women's club. Most will offer playgroups - that's what you're looking for. They'll offer playgroups with kids of different ages, so you can meet with 4-6 other moms regularly that have 1 year olds. Playgroups, and the larger groups alike, will (or should) set up monthly outings to places like the park, zoo, train ride, etc. It's a great way to explore new areas with other kids and moms. They'll also have a meeting once a month with a speaker... someone who usually discusses a topic relevant to moms with babies, toddlers, and preschoolers.
In terms of one-on-one time, the chalk and crayons are a great idea. Here are some others:
PLAYDOH (be sure to get cookie cutters, rolling pin, etc. - and a plastic mat for easy clean up)
PARK, CREEK, DUCK POND, etc. We used to do this several times a week. Check out new parks in your area - we used to frequent about 6 different parks - a new outing every time. Kids love it and it's a great way to wear them out.
LIBRARY STORYTIME (some bookstores offer this as well)
EXPLORE THE MALL. Get there early before it's too crowded and let him walk beside you. It will really wear him out, and you wouldn't believe the things they'll discover while they're there. Grapevine Mills is a great one with the allagator, ramps, etc. One lap is one mile.
VISIT BASS PRO in Grapevine. They've got a couple of fish tanks, a bridge and koi pond, and lots of cool lifelike animals.
VISIT GAYLORD TEXAN (parking is $10, so you might want to save this for a family outing). Again, a great indoor place to explore.
SPORTS Even the youngest kids, as long as they can run, appreciate kicking around a soccer ball in the backyard or playing catch with mom with a spongie nerf ball.
FOUNDERS PLAZA - Just north of DFW Airport, you'll find their observation area. There area picnic tables, telescopes, etc. Sounds from the tower are also piped in to the area, so you can hear what's happening between pilots and the controllers.
GRAPEVINE REC CENTER - On Thursdays they offer "Play Time" for toddlers and preschoolers in their gym for $1 - $2 (I don't remember). They bring out balls, games, mats, and the large parachute for kids to play with.
There are also lots of classes and activities that you can sign your child up for. Some are "Mommy & Me" others children participate and moms watch. I would start with your local rec center (through your city parks & rec dept.), then also check out places like Little Gym, Gymboree, etc.
www.familyeguide.com Free guide to affordable family activities in Lewisville, Flower Mound, Highand Village, and the surrounding communities.