Well done.
I negotiate for a living. These are my tips for any negotiation:
1. Assess your bargaining position independently, and use that to form a bottom line.
2. Always volunteer to put the first figure out there. The disadvantage (of offering up "free information") is far outweighed by the advantage of anchoring the conversation.
3. Put your first offer comfortably high, to give yourself room to come down to a bottom line.
4. Never say something is your bottom line unless you mean it. If you're in a grey zone, give yourself a get-out clause ("Unless circumstances change I can't see myself moving from my present offer") that affords you flexibility to either hold out or move a little. Situations change so don't box yourself in unnecessarily.
5. Don't negotiate unless you have the ability to walk away. If you can't walk away, you are not negotiating. You are begging.
6. Usually the party that gets to its bottom line "first" (and holds it) has more success in a negotiation. You want the last step to be the other party coming up a small amount to your figure, rather than the opposite.
7. Where possible, only negotiate where you have something or someone to "blame" (i.e. a competing offer, an alternative strategy or someone else involved - e.g. your partner). That allows you to put the blame, or rationale, for your position on the "external blame sponge", therefore defusing the present conflict.
8. Listen to what the other party says when making/receiving offers. He, she, it or they will usually give you some indication of the relative strength of his, her, its or their position.