Map from Google - Serbia

In Serbia, the climate is continental, with cold winters and hot summers. The average temperatures in the plains are around 0 °C (32 °F) or slightly higher in January, and around 22 °C (72 °F) in July.
Here are the average temperatures of the capital Belgrade.
Average temperatures - Belgrade
BelgradeJanFebMarAprMayJunJulAugSepOctNovDec
BelgradeJFMAMJJASOND
Min (°C)-20381215161613840
Max (°C)461218232527272418115
Min (°F)283237465459616155463932
Max (°F)394354647377818175645241

The country can be affected by cold air masses of Siberian origin, as well as by warm air masses from the Mediterranean or even Africa, so the temperature can vary greatly depending on the weather situation.
The climate in the area is fairly homogeneous: in southern Serbia and Kosovo (which is not yet clear whether it will become an autonomous region or an independent state), the lower latitude is offset by the higher altitude (being an hilly area), therefore here the temperature can be even lower than in the rest of the country: in Priština, located at 575 metres (1,900 feet) above sea level, the average of January is -1°C (30 °F), that of July is 20 °C (68 °F), and nights remain quite cool even in summer.
Here are the average temperatures of Priština.
Average temperatures - Pristina
PristinaJanFebMarAprMayJunJulAugSepOctNovDec
PristinaJFMAMJJASOND
Min (°C)-5-3049111312951-3
Max (°C)261116212426272317104
Min (°F)232732394852555448413427
Max (°F)364352617075798173635039

Only in the south-western mountainous area, which reaches 1,500 metres (5,000 feet), the climate becomes much colder, with freezing and snowy winters, and cool summers.

Precipitation follows different patterns depending on area, but June is quite rainy in almost all of the country, mainly because of afternoon thunderstorms, which may occur from May, reach a peak in June, and then decrease in July and August, without disappearing completely until the end of summer.
For the rest, the western regions receive generally more rainfall than the eastern ones; in the former, the months of November, December and January are quite rainy (or snowy) as well.
Annual precipitation drops below 600 millimetres (23.5 inches) in northern Vojvodina, on the border with Hungary and Romania; it is around 600 mm (23.5 in) in eastern Serbia, on the border with Romania and Bulgaria, and in the south; it's between 600 and 700 mm (23 and 27 in) in central Serbia and in Belgrade; and finally it exceeds 700 mm (27 in) in western Serbia.
Here is the average precipitation in Belgrade.
Average precipitation - Belgrade
BelgradeJanFebMarAprMayJunJulAugSepOctNovDecYear
BelgradeJFMAMJJASONDY
Prec. (mm)454545557595605550556055695
Prec.(in)1.81.81.82.233.72.42.222.22.42.227.4
Days98789855479988

In Kosovo, annual precipitation is around 800 mm (31.5 in) in the western part (Pec, Prizren), while it's considerably lower in the eastern part (Mitrovica, Priština), where it's around 550/600 mm (22/24 in).
Here is the average precipitation in Pristina.
Average precipitation - Pristina
PristinaJanFebMarAprMayJunJulAugSepOctNovDecYear
PristinaJFMAMJJASONDY
Prec. (mm)403540507060504540457055600
Prec.(in)1.61.41.622.82.421.81.61.82.82.223.6
Days14121112131288891215133

In mountain regions, more rainy in summer and snowy in winter, precipitation exceeds even 1,000 mm (40 in) per year.
August and September are typically the months with the lowest number of rainy days.
In winter, even where the amount of precipitation is not high, the frequency is still fairly high, which means that there are frequent snowfalls, though typically not abundant.

Belgrade

Winter, from December to February, is cold. Some periods are cloudy or foggy, with temperatures around freezing (0 °C or 32 °F), while in other period, cold winds of Arctic or Russian origin prevail, and they can break out violently in the country, followed by intense frosts, when cold air settles on the ground. In these situations, the minimum temperature may drop to about -20 °C (-4 °F).
On other occasions, mild air masses from the Mediterranean can penetrate in the country: in these cases, the daytime temperature can exceed 10 °C (50 °F), especially in the centre and south, and more rarely in the north.

Spring is an unstable season, characterized by mild periods alternating with the return of cold weather, in which late snowfalls can occur, in March and sometimes even in April.

Summer, from June to August, is warm, with a wide daily temperature range, and generally with cool nights. In fact, lows are often around 14/15 °C, and highs a bit below 30 °C (86 °F). But at times, hot air masses from Africa can reach the country, and due to continentality the heat becomes oppressive, with temperatures that can exceed 40 °C (104 °F); however, usually these periods do not last long.

Autumn is not very rainy, because the bulk of the moisture carried by Atlantic frontal systems is deposited as precipitation on the western side, in the countries overlooking the Adriatic Sea. Only in the westernmost area, autumn becomes rainy, at least in November.

When to go

The best time to visit Serbia and Kosovo is from May to September. In May and June, afternoon thunderstorms are possible, and less frequently also in July and August, in which in return there can be hot days. In general, nights are cool, except during the most intense heat waves. Those who don't like the heat, can avoid July and August. Therefore, the month of September, usually sunny and pleasantly warm, may be interesting.

What to pack

In winter: warm clothes, sweater or fleece, down jacket, hat, gloves and scarf, raincoat or umbrella.

In summer: light clothes, T-shirts and shorts, but also long pants, light jacket and sweatshirt for the evening; raincoat or umbrella.