My DD never would take a bottle... she did like the sippy cups that don't have the regulating thing in them, though... She essentially went straight from breast to sippy cup. And she's not the only child I've met who did this.
So, if she's willing to take the sippy cup, go with it!
As for the solids thing... the American Academy of Pediatrics says exclusive breastmilk or formula for 6 months... but many pediatricians don't follows those guidelines and encourage moms to start giving solids at 6 months. There are specific things you want to be looking for in your baby before you give solids. Here's a great online article from LLL about starting solids: