Nothing wrong with your first race being a 10K.  Once you get to 3 miles, just add a mile per week to your long run, to get to 6 (or 7 if you have time).  My first race was a 10K and I know I hadn't run longer than 4 miles in training. (I wasn't training to race but a friend asked me to run the 10K as part of a team.)  When I started running races, 5Ks were rare: the standard short race was 10K  I did a half-marathon and a 10-miler before I did a 5K. My first 5K was late in my second year of racing.

Len

Years ago, before things like the internet and forums, the 10k was king of the recreational races and it was the race to run for first timers and experienced runners alike. Pay attention to your body as you train and don't take things too fast but I don't see any reason why you couldn't do a 10k before too long. Move forth and register.

It is so true that the 10K was the race when I started running races in 1987.

I had run five 10K's, a 15K, and a marathon before I ran my first 5K.

I always remember finishing the first 5K and saying to myself at the finish "That's it?"