The classical sonata form has both the mandatory key change and the sections. In the romantic era, the sections are very often also marked by a change in mood and/or tempo.
This form is virtually always used as basis for the first movement of symphonies, it is also popular for chamber ensemble pieces like string quartets.
The classical sonata starts with an initial theme, and after stating that, modulates, usually to the dominant key. Then a second, contrasting theme is introduced. This sequence of first theme / second theme is called the exposition. In many cases this is repeated once almost completely.
After the exposition section there is a development section. This may itself be divided in separate sections, which are often marked by a key and/or tempo change.
In many cases a clear reprise or recapitulation section may be identified that closes the movement.
Here's an example:
(Brahms, string quartet a minor, movement 1)
You can clearly spot the sections since they have double barlines and labels (A, B etc)
Another, completely different type of form that also has a sectioned design as well as the mandatory key change is the fugue.
The fugue starts with a single theme in one voice. After the first statement of the theme, a second voice answers with a re-statement of the theme; almost always a fifth higher or a quarter lower. In many, many cases, the answer actually establishes a change of key. Typically the answer will be followed by one or two (sometimes more) entries, always alternating theme and answer until all voices have entered the piece, concluding the exposition section.
The rest of the fugue is a sequence of sections. Sections are either episodes, using mostly free counterpoint, or a sequence of some motive from the theme or developmental sections that restate the theme, often in a transformed form, or using some kind of canonic treatment of the theme (like stretto). Sections boundaries often coincide with a change of key or mood (although an explicit change of key is not generally denoted in the score - it is solved using accidentals)
A fugue will generally end with a reprisal section, restating the original theme in the original key.
Fugues will generally not write explicit key changes, and will typically not have explicit tempo changes. However, the character of the sections may suggest a different pace or mood.
Here's a fugue that has a section that suggests double time:
(Bach, WTC fugue 4)
About half way, there's a section that has a line with a lot of eights suggesting double time; this really stands at as compared to the stern and grave theme that starts of the fugue.