Small wheel bike to be converted to bigger wheels
Cheapest, safest and most practical option option would be to sell and buy  a second hand bike. 

For instance I just sold and old, but perfectly usable full suspension MTB that originally cost about $US3500 in 1995 for $US60.00. The buyer got a bargain and I got some space back in my garage. Look for someone who wants some garage space back.... 

If however you are wanting to make a franken-bike for fun, you will also need to consider the increased wheel size will probably need different gearing, and the frame will be subjected to quite different forces than what it was designed for. You would need new chain, new wheels and tires. Brakes may not be in the right places and might not fit. As it was a kids bike the cranks might be too short.