As was brought up by Ariah, a true barter system has never really been observed, even in very undeveloped parts of the world. People tend to implement the closest things to barter systems in time of need (winter, famine, etc.), or when communities are extremely small. Often, however, it's usually more of a neighborly, gift-based system.
Now, you have stated that you want your community to move past a communal, gift-based, bartering society, which is really quite natural. I can see it going in two directions, one without money, and one with money. In this case, I determine money to mean a medium of exchange that holds "imaginary" value. A banknote is not actually worth 20 dollars - its literally a piece of paper - but the society gives it value.
Now, in a stable environment, often several goods naturally become a sort of currency among a people. They generally meet a few criteria:
- Non-perishable (I use this term loosely)
- Stable in value
Depending on the environment, a few common examples would be cattle, rice, blankets, salt, skins, tobacco, and wine. All of these never really lose their value (okay, cattle can die, but they're pretty stable), are useful, and, in the areas where they were used as currency, were generally pretty easy to come across. Things like bread, or eggs, that will go bad extremely quickly, are less suited to this system, and things that fluctuate greatly in value (because of seasons, for example), are generally unsuitable as well.
This would be a more natural evolution as the society moves on from struggling to survive to a more normal state of existence. Without external influences, this is the most likely development to occur.
Well, we all know how money works. As for materials, you can be as creative as you want. Whether its a seal printed on paper, minted coins, bits of precious metals, shells, et cetera, you can really choose anything convenient for use as your currency.
"Why would such a system develop here though?", is the imperative question. Frankly, it wouldn't be completely necessary. It is quite plausible, however, mostly due to the cultural background of the settlers. If they came from a nation with ideas of money engrained in them, it is quite likely that they would mimic their native culture.
Somewhere in Between
Most European colonists (which seems to be the closest approximation to yours), used a mix of the two. In early Virginia (Jamestown and Roanoke), copper, tobacco, and minted coin were all used as currencies. Most emerging societies will find themselves in between as well.
Money, Credit, and Banking in Virginia, 1585-1645
Establishing a Money-Based Economy
Establishing a money-based economy is fairly simple in theory. The government decides upon a value to give to the currency and makes the currency. The hard part comes in convincing everyone to act in accordance with the idea. It could be difficult to convince someone to trade a cow for a piece of paper at the start. As time goes on, however, and if the money stays stable (no horrible inflation or deflation occurs), the society will start to have more and more confidence in it. It certainly will help that all of the settlers came from a society where money was the norm. Money, at the end of the day, is a risk, with rewards. It has no inherent value, but allows for much more ease in trade and regulating the economy.