The rules for punctuation around transitions and connectors are brief.  There are, basically, three ways to punctuate transitions and connectors within essays.  Note the following examples:

I studied hard for the exam. However, I failed.  (Capital letter following a period)
I studied hard for the exam; however, I failed.  (Lower case letter following a semicolon)

The weather in Alaska is cold in the winter. Otherwise, I like living there.   (Capital letter following a period)
The weather in Alaska is cold in the winter; otherwise, I like living there.  (Lower case letter following a semicolon)

The punctuation in both sets of examples above is identical in terms of "correctness." Neither the use of the semicolon or period is "preferable" to the other. Probably the use of one or the other will be influenced by the sentences around it, with "sentence variety" a factor in choosing which to use.

Less frequently, transitions and connectors are used within a sentence as follows:

The president promised a quick victory. Victory, however, was not easily won.
Most people think the senator was honest. He is, instead, a thief and a liar.

Note that in the sentences above, the transition is set of by commas on both sides.  Also note that it would be possible to remove both the commas and the transitions without affecting the "correctness" of the remaining sentence.

Summary:  In sum, transitions and connectors show the relationship between ideas within sentences.  In addition, correctly placed and punctuated transitions and connectors can help guide the reader through a complex maze of ideas.

The quiz, Punctuation Around Transitions and Connectors, is HERE (will open in new window).