The earliest stringed instruments for which we have physical evidence are lyres from early ancient Mesopotamian states (~2500 BCE). Archaeologists have excavated a few fragments that can be positively identified as such. Egyptian tomb paintings and statuary from even earlier (~3000 BCE) demonstrate that harps, lutes, and lyres of various sizes and shapes have existed in more-or-less the same forms as they do now for basically all of recorded human history. Here you can find a website detailing a guy's efforts to reconstruct a New Kingdom-era arched harp, along with some reproductions of contemporary wall paintings (dating from about 1000-1500 years later than the earliest ones, which are much more difficult to track down). We've found bone and clay flutes that date back tens of thousands of years, so it's not totally unreasonable to presume that stringed instruments (and instruments from other families, as well) also existed alongside them in some form, but simply aren't physically extant any longer.
Bowed strings, the ultimate ancestors of the modern violin family (violins, violas, and cellos)*, entered Europe from the Middle East in the early Medieval period, and were developments of the spike fiddle type that became common in Turkey and Arabia no later than the 8th century. The most common European iteration of this type was the rebec, which entered eastern Europe by the 9th century and proliferated very quickly. The rebec remained in use until deep into the Renaissance era.
Plucked strings with a parallel sounding body (rather than a perpendicular one, as in harps and lyres) were known in Europe by the 12th century, again derived from a Middle Eastern instrument, the lute (al-oud in Arabic, hence the word "lute"). From these, we get the modern guitar and its variants.
The two previous paragraphs focus primarily on the evolution of European stringed instruments. Other cultures around the world, sub-Saharan Africa, and the Far East most notably, have their own traditions, which developed more-or-less independently from what we presume are common origins. Indian instruments (sitar, sarod, etc.) are a separate branch of the same family of Middle Eastern instruments that preceded the European instruments.
* Interesting side note on this: the origin of the modern orchestral double bass is muddy and frequently contentious, but it has some elements that are more in keeping with the viol family or the viola da gamba than with the true violins. Those commonalities include its sloped rather than rounded shoulders and the fact that the strings of the double bass are tuned in fourths, rather than the fifths tuning of the violin, viola, and cello. Viols, once the primary bowed stringed instrument in both privileged and popular European music, fell out of favor in the late 16th century and was almost completely replaced by instruments of the violin family.
Another interesting side note: the full name of the cello is actually violoncello, which means "little violone". The violone was the largest and lowest-pitched instrument in the viol family.
