Whether or not you're shopping for a diamond, the four Cs are a good place to begin. Some or all of them can apply to other gemstones as well. Precious gems (diamonds, rubies, sapphires, and emeralds) are typically priced by carat weight, so obviously a larger stone will be pricier. Carats are a measurement of weight, not size. Some stones are less dense, and so weigh less than a similarly sized stone of a different type. Larger carat sizes are also much rarer, so the price can rise exponentially.
The next C to consider is cut. This refers not only to the shape of the stone (read more here), but also the proportions of how it is cut. If it is too deep, it will weigh more (and thus cost more) but not reflect light as well as it could. A shallow stone may appear large for its weight, but it will also reflect less light and may have a thin girdle, or edge, that is prone to chipping. Cut shape is very much a personal choice. Round is still the most popular, but princess, oval, pear, cushion, and more can be found in engagement settings.
Color, perhaps surprisingly, is used for diamonds to describe their level of colorlessness. The most common scale is the GIA's, which is considered the industry standard. They describe color in letters, from D to Z. Most diamonds have a slight tint of yellow, brown, or grey to them. The average non-gemologist person often will not be able to notice these tints in any diamond that is graded about I or higher, while D-F are considered the 'colorless' grades. Beyond Z are what are called "fancy" color diamonds. These include all shades of blue, red, green, pink, purple, orange, black, and intense shades of brown and yellow.
Clarity often has the largest impact on price after carat. Again, GIA's scale is the standard. It ranges from Flawless (F and IF for 'internally flawless') down to Included (I1, I2, and I3). VS1 and VS2 (Very Slightly Included) are good choices for a balance between price and beauty because the inclusions are not visible without magnification. Even Slightly Included (SI1 and SI2) can provide good value since the inclusions may not be very easily seen with the naked eye.
Carats are a measurement of weight, not size.
And you can start by throwing that 'three months salary' line out the window. It's nothing more than a marketing scheme. Consider several factors: what you, personally, can comfortably afford; what characteristics your partner wants in a ring that will affect price (metal, carat weight, center gemstone, etc.); and financing options. Many companies can provide convenient payment plans for your purchase, including PayPal! Look for deals that offer a period of low or no interest and calculate whether you can complete the payment in that time to save money.
More and more couples are selecting the engagement ring(s) together. This is a great way to ensure that your partner gets exactly what they want. It also gives you amazing insight into their taste in jewelry for future surprise gifts and takes the pressure and worrying out of the proposal. Even if you want the final choice to be a secret, browsing with your future spouse is an excellent option for making sure you're both on the same page. This should not feel like a test you might fail but an affirmation that you are building a future together.