天気
This is the most common word for weather, and expresses a naive concept of the entire perceptive state of sky (and air) in some place at some moment.
It includes sunshine, cloudage, precipitation, wind, humidity and temperature, but not likely air pressure.
By at some moment I mean, this word is expected to state an overall impression at a certain time-point, or in a short timespan, which has no complex changes of state needed to mention.
You can assume a unit of 天気 to be a piece of information that a single column of a table in the newspaper's forecasting section can describe.
On actual usage of the word 「天気」
○ 良い天気 天気が良いです。
○ 晴れの天気 天気は晴れです。
？ 晴れている天気 天気(は/が)晴れています。
× 寒い天気 天気(は/が)寒いです。
○ 死ぬほど寒い天気 天気(は/が)死ぬほど寒いです。
△ 涼しい天気 天気(は/が)涼しいです。
If you'd like to use the word 天気 itself in a sentence, you should take care that most expressions people like to use with 天気 are subjective evaluations (row #1) or metaphors.
Only this sort of expressions can used with 天気 without limitation.
When you need a word that means a part of 天気 (be sunny, rainy, windy etc.) to explain 天気, don't use it adjectively (#3) but appositively (#2). 天気 usually rejects such "a round circle" type attributions.
Though, in many cases, it's enough to mention them without 天気: 晴れています "It's sunny."
Don't say 暑い天気 and 寒い天気, because the two words already contain "feel hot/cold in environment" senses (#4).
You can however put your comments to make them subjective statements: 寒すぎる or 死ぬほど寒い (#5).
Other objective words also generally don't get along with 天気, for example, 涼しい天気 is only valid when it means "comfortably cool".
天候
It also refers to the state of atmosphere observed in a place, but this one mainly describes successive shift or tendency in a period of time.
In a short term whose time lapse can be ignored, it becomes interchangeable with 天気.
Since this word is an unnativized Sino-Japanese compound carries formal mood, we don't use it much in casual conversation. If I say "the weather turns bad" with 天気, I prefer 天気が悪くなる, but with 天候, 天候が悪化する.
I expect from この一週間の天気 the weathers of Sunday, Monday, Tuesday..., but from この一週間の天候 a description of how weather changes, getting warmer or colder, whether typhoons come or not, and so on.
Other usage is basically the same as that of 天気.
気象
This could be translated as meteorological phenomenon or "meteorologies", but not very common to be used as noun (as in 異常気象 "abnormal meteorology").
The main usage of it is as adjective, meteorological (as in 気象衛星 "meteorological satellite").
This word refers to each physical phenomenon detached or not observable from a single viewpoint.
As you can see, satellites only monitor 気象 and not 天気 or 天候 (unless you say that the solar wind or the magnetosphere oscillation is 地球の天気).
You can't mention El Niño and La Niña as 天気 or 天候, either.
気候
Basically identical to climate, that is a unique environmental feature of a location never changes overnight.
You want to live in Florida because 気候が良い, to go for a trip to Florida not Louisiana because 天候が良い, and to choose to go to FL not LA tomorrow because 天気が良い.
