It is neither a strawberry or a begonia. I do agree that it looks like a combination of both. Strawberry Begonia is Saxifraga sarmentosa or S. stolonifera. There are a couple common names Mother of Thousands or Strawberry Geranium. The foliage is dark green with silver veins covered with hairs. The underside is red or purplish. The flower are small on tall stems above the foliage. There is a variegated form called 'Tricolor"  that has a cream colored edge around the edge. It slows slower but is more attractive. It is also slightly higher priced.

Water: This plant needs constant source of moisture so should not be allowed to dry too long. If they roots stay dry  they can die.

Fertilizer: A balanced fertilizer is best since the flower are not important. I like Jack's Classic 20-20-20 but any balanced fertilizer will work.

Soil: This is a fibrous rooted plant so likes a evenly moist soil. This means it can't stand sitting in water for days or being bone dry. It will grow in soil from your yard but would do well in something similar to African violets.

Light: It needs to be protected from full sun. A eastern or northern window is best.

Temperature: The plant like to be on the cooler side 55 - 65 is best. I would not recommend it for 90+ weather.

Humidity: This plant enjoys humidity provided the air is circulating. It can depend powdery mildew in moist but slow moving situations.

Ground Cover: It can be grow outside but needs a shady protected spot. Direct sun will burn the leaves. In mild climates it will grow back from the roots over winter. The plants are short and easily over shaded by larger plants.

Propagation: This is plant is nicknamed  "Mother of Thousands". It is constantly produces stolons. Stolons are little plants forms at end of stem. Other plants that do the same thing are Strawberries, Episcias, and Spider Plants. Just place a plantlet or stolon is a pot of soil. You can use a bobby pins or similar item to hold it in place. It should roots within a few weeks. When it has rooted cut the connection and you have a new plant. Outside it will form runners but needs bare soils to root.

Sources: You can find in various shops or stores. Also it is available on eBay.