Skillet and frying pan are fairly interchangeable - generally a broad (usually around 30cm), circular pan with slightly sloped raised sides around 2-4cm high.
Saucepans are much deeper in relation to their base width, however. Saucepans come in a variety of sizes, with small pans generally being about 15cm across and around 10cm high, all the way up to big stock pots that can hold 3 or 4 litres of liquid that will be a good 30cm tall if not more. Saucepans almost always come with a lid.
Frying pans, as the name suggests, are made for shallow frying food. Saucepans are made for making sauces and cooking things in liquid. They are better suited to this as the higher sides and narrower base, along with the lid, serves to give better control of evaporation.
Boiling vegetables in a frying pan, with a large surface area to evaporate off and no lid to keep the liquid in, will mean you will either burn your vegetables or constantly be topping off the pan with hot water.
If you are going to be cooking small amounts of vegetables regularly I would suggest you invest in a smaller saucepan, or a set of three different sizes to give you some options.