You won't be able to play as a summoner until you get a job to Lv30 and complete the required quest to unlock SMN.
I suggest levelling BLM first, probably just to 18 for now, then WHM to 30 or whatever to sub for SMN, and then go get SMN and unlock that.
Also, keep in mind that you CAN summon everything at Lv1, but you have to and actually get it. The only avatar you start with is Carbuncle. All of the other avatars must be defeated in order to be able to summon. There's two different fights you can do for most of them: A Lv20 solo fight, or a Lv60-ish full party battle. The only exceptions are Diabolos and Fenrir, which both require Lv70ish I believe. Diablolos also requires you to have completed the Chains of Promathia missions.'
There's also elemental spirits, which can be purchased at the auction house for varying prices (depends on the spirit. Earth spirit is pretty cheap, but stuff like light and dark spirit cost over a million gil) and are obtained other ways as well (Light and Dark are so expensive because they drop off a high level BCNM battle. That's a Burning Circle Notorious Monster, which is a timed, (usually) level-capped battle against difficult enemies in a special battlefield.)
Please, please, PLEASE sub WHM when you play SMN. It gives you more MP (which you will NEED) and the ability to heal, which helps your party greatly. /THF adds little of value but Treasure Hunter, which is only useful for farming, which you will to solo. /DRG adds absolutely nothing you'll need for SMN. You can't use your wyvern with /DRG, only DRG main.