How can I get the list of a columns in a table for a SQLite database?

Here's the simple way:

.schema <table>

What you're looking for is called the data dictionary. In sqlite a list of all tables can be found by querying sqlite_master table (or view?)

sqlite> create table people (first_name varchar, last_name varchar, email_address varchar);
sqlite> select * from sqlite_master;
table|people|people|2|CREATE TABLE people (first_name varchar, last_name varchar, email_address varchar)

To get column information you can use the pragma table_info(table_name) statement:

sqlite> pragma table_info(people);
0|first_name|varchar|0||0
1|last_name|varchar|0||0
2|email_address|varchar|0||0

For more information on the pragma statements, see the documentation.

The question is old but the following hasn't been mentioned yet.

Another convenient way in many cases is to turn headers on by:

sqlite> .headers on

Then,

sqlite> SELECT ... FROM table

will display a headline showing all selected fields (all if you SELECT *) at the top of the output.

Here's a SELECT statement that lists all tables and columns in the current database:

SELECT m.name as tableName, 
       p.name as columnName
FROM sqlite_master m
left outer join pragma_table_info((m.name)) p
     on m.name <> p.name
order by tableName, columnName
;