Map from Google - Izmir

In Izmir (or Smyrna), big Turkish city on the Aegean Sea, the climate is Mediterranean, with relatively mild winters, and hot and sunny summers.
Here are the average temperatures.
Average temperatures - Izmir
IzmirJanFebMarAprMayJunJulAugSepOctNovDec
IzmirJFMAMJJASOND
Min (°C)66711151922221814118
Max (°C)121316212631333329241814
Min (°F)434345525966727264575246
Max (°F)545561707988919184756457

Precipitation amounts to 700 millimetres (27.5 inches) per year, and is most abundant in winter, while in summer it rains very rarely.
Here is the average precipitation.
Average precipitation - Izmir
IzmirJanFebMarAprMayJunJulAugSepOctNovDecYear
IzmirJFMAMJJASONDY
Prec. (mm)1351057045401032155085140700
Prec. (in)5.34.12.81.81.60.40.10.10.623.35.527.6
Days1087542002461159

Winter, from December to mid-March, is relatively mild and quite rainy: the average temperature in January is 7 °C (44.5 °F), and it rains on average for about 10 days a month. The city is less exposed to cold waves than Istanbul, however, cold air masses can still arrive, leading to night frosts, for example in January 2013 the temperature dropped to -8 °C (18 °F).
Snow is rare, but not impossible: for example, it snowed in February 2012.

Spring, from mid-March to May, is very mild, and while in March it's still a bit unstable, later it gradually becomes sunnier, and with less rainfall.

Summer, from June to mid-September, is hot, dry and sunny. Typically the humidity is not very high, and sea breezes blow, in addition an insistent wind from the north, the Meltemi, can blow; however, there may be periods of intense heat, with highs about 35/38 °C (95/100 °F), and even 40 °C (104 °F).

Autumn, from mid-September to November, is pleasantly warm and sunny at the beginning, while rainy periods become more frequent over the weeks; however, October is often still a pleasant and sunny month, especially in the first half.

The amount of sunshine in Izmir is great in summer, when the sky is practically always clear, while in winter, sunny periods alternate with periods of bad weather, with cloudy skies.

Izmir

In Izmir the sea is never very warm, because in summer the north wind prevents it from warming up much, however, the sea temperature reaches 22/23 °C (72/73 °F) from June to September.
Here are the average sea temperatures.
Sea temperature - Izmir
IzmirJanFebMarAprMayJunJulAugSepOctNovDec
IzmirJFMAMJJASOND
Sea (°C)151515161822232322201816
Sea (°F)595959616472737372686461

When to go

The best times to visit Izmir, for exploring the city, the monuments and the neighbouring towns such as Ephesus and Pergamum, are spring and autumn, ie the months of April and May, and from mid-September to mid-October. In fact summer is often hot, especially in July and August.
For beach life, you can go from June to September.
September is probably the best compromise between beach life and cultural tourism, if you want to do both.