One on one interaction is important, as with any child. They just soak it up more than most. Even hum-drum activities can be educational. For instance, when we drive in the car on errands we'd recite every nursery rhyme and poem we could. I'd start it, they'd finish. We'd count by 5's, 10's, 2's, 3's, etc. Sing every song I could remember (not just kid songs), then play categories (ask them all the types of trees they know, flowers, types of music, cars, etc.) It all requires so much energy on Mom (or Dad's) part, but they are constantly thinking.
Probably my most important advice is reading to them (until they lose patience with it--when they can plow through books faster than your voice holds out). Read fiction, non-fiction, poetry, sayings, etc.
When you're just plain beat, educational computer games are great to have them work on when you need a quick nap--they might need you there at first but will quickly get the hang of it. Read as much as you can yourself about giftedness. There are some great articles and books out there.
Consider testing young (4? or 5?) if your child is very precocious. If your toddler is reading, see about early acceptance to kindergarten, or skipping kindergarten and going to first grade.
I agree with Trinity about exposing them to all kinds of things. Museums, workshops, ceramics classes, etc. play board games that are designed for older children. Prepare yourself for a lot of work and explaining everything. I used to make cookies all the time with my 2-3 yr old and would explain about measurements, nutritional info (food groups), chemistry, etc. while doing so.
I also agree that there is a financial legacy that goes with giftedness in terms of giving them opportunities (lessons, books, online subscriptions, magazines, software, games, and later camps and distance learning opportunities...before you even get to college).
Finding playmates is a challenge. My kids luckily play together (I have 4) even though the oldest is 7 years older than the youngest. They prefer it to age-mates. Highly gifted only children probably depend more on their parents for interaction.
GOOD LUCK!
