you can start training summoning without completing the quest

A good option is to start by doing the Wolf Whistle quest since you will reach a summoning level of 4 after it. However, you can start training summoning without completing the quest.

For many centuries now, the druids of Taverley and some of the ogre shamans of Gu’Tanoth have been bringing various familiars from other realms to help them with a variety of their tasks. Of course, the druids being peaceful and the ogres being more ‘business-like’, they traditionally summon entirely different familiars – what would a druid want with a rune minotaur, for example? What would an ogre do with a unicorn stallion?
Having realised how useful this skill is to the adventurers of RuneScape, they have released the secret of their powers, and now you can enjoy the curious advantages that come with learning the art of Summoning.

Summoning works by ‘infusing’ the essence of a creature from a strange realm into a type of pouch. When the Summoning pouch is opened, a portal is briefly activated to pull the familiar through to the world of RuneScape, where it is bound to serve its master for a period of time before the portal closes and it is pulled back.

Summoning: The Basics Game Guide
It has been revealed by Pikkupstix, when you reach Level 99 Summoning, that the place where your Familiars hail from has never been part of Gielinor, yet is nevertheless part of our lives every day. Familiarisation allows you to access this realm, known as the Spirit Plane, for limited amounts of time, where you and a Familiar can hunt for raw shards. Other races than the druids, ogres, and now the humans, appear to possess some knowledge of Summoning; for instance, the Elf Warrior drops charms, hinting that the art is known to the Elven race as well.