First the store FAQ:
And the highlights on Trading:
Q: Will I be able to trade items?
A: Yes. We've added the long-requested Trading feature, so you can swap items with friends.
Q: Are all items tradeable?
A: No, not all items are tradeable. Some limited-availability items are not tradeable. Purchased items are also not tradeable at the moment (see About the Store).
Q: Can I just give items to someone? Or do they have to give me something back?
A: Yes, you can "gift" items to others. They do not have to give you anything back.
Q: Can I craft the Polycount weapons? Do they have recipes like the other community items?
A: Yes, they're craftable. They have the same recipe style (combine two weapons) as the other community-contributed items.
So, store bought stuff not tradeable. And trades don't have to be trades, they can be gifts.
Post what you are willing to trade/give, what you want to trade/be gifted, etc. Try to keep things on a first come first served basis for responses. Radical Ans has also made a spreadsheet that might make things easier, with have, wanted, and offer columns. If you see a deal you can work from someone you can write in the offer column.
Also here are the new recipes for this update. You might want to consider crafting your spare weapons into the new ones before trading them away:
Mad Milk = Jarate + Reclaimed Metal
Shortstop = Force-A-Nature + Reclaimed Metal
Holy Mackerel = Sandman + Reclaimed Metal
Black Box = Direct Hit + Reclaimed Metal
Battalion's Backup = Buff Banner + Reclaimed Metal
Degreaser = Backburner + Reclaimed Metal
Powerjack = Axtinguisher + Reclaimed Metal
Sidney Sleeper = Huntsman + Reclaimed Metal
Darwin's Danger Shield = Razorback + Reclaimed Metal
Bushwacka = Eyelander + Reclaimed Metal
L'Etranger = Dead Ringer + Reclaimed Metal
Eternal Reward = Cloak & Dagger + Reclaimed Metal