How do I turn on abbrevs by default just in mode mymode?

Abbrev mode expands abbreviations as you type them. To turn it on in a specific buffer, use M-x abbrev-mode. To turn it on in every buffer by default, put this in your init file (see How do I set up an init file properly?):

(setq-default abbrev-mode t)

To turn it on in a specific mode, use:

(add-hook 'mymode-mode-hook
          (lambda ()
           (setq abbrev-mode t)))