The cheapest way to buy a 125cc moped is to buy a used one directly from the owner. They can be found through classified advertisements online or in the newspaper, as well as auction sites such as eBay.com
Buying a cheap 125cc moped new is a matter of comparing prices at various dealerships. Different models are also different prices. For example, a well-known brand such as Vespa is generally more expensive than a smaller, less in-demand brand. Optional add-ons such as cargo racks can also increase the cost.
When comparing dealer prices, be sure to factor in any warranties or service contracts. Mopeds generally need routine scheduled services in the first year or so of use as the engines are run in. This can significantly add to the cost of the moped, so if one dealer includes a special deal on those services but charges a slightly higher initial price, it may work out to be cheaper in the long run.
Used mopeds can be found just like any other used vehicle. Some dealers sell them, which may be a good choice because it gives the benefits of a dealer while still being less expensive in the long run. Searching for "used 125cc mopeds" on a search engine can bring up moped-specific websites. If buying a preowned model, it is best to try and find a local one, so it can be inspected and test ridden before purchase.