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In addition to the advancement by your character level, I assume since it says
Animal Companion (Ex): You gain an animal companion. Your effective druid level for this ability is equal to your sorcerer level – 3 (minimum 1st). This bloodline power counts as your bloodline arcana and also replaces laughing touch.
that you look to the animal companion entry at druid.
Now considering this, does it advance in levels as you advance, ie at character level 10, you'd look at the chart and apply lvl 7, which gives it 6hd +4BAB +5/+5/+2 saves 6skills, 3feats +4ac +2str/dex 3 tricks, and evasion, devotion, etc.
Assuming you continue to level up in sorcerer, yes, your animal companion will advance as if you were a druid of your sorc level-3.
So at 10, your companion will indeed be the same as that of a level 7 druid, and at 20, it'll be like a level 17 druid's companion. You've got the right idea.
