What is the equivalent PostgreSQL syntax to Oracle's CONNECT BY ... START WITH?

Use a RECURSIVE CTE in Postgres:

WITH RECURSIVE cte AS (
   SELECT key, value, 1 AS level
   FROM   taxonomy
   WHERE  key = 0

   UNION  ALL
   SELECT t.key, t.value, c.level + 1
   FROM   cte      c
   JOIN   taxonomy t ON t.taxHier = c.key
   )
SELECT value
FROM   cte
ORDER  BY level;

Details and links to documentation in my previous answer:

• Does PostgreSQL have a pseudo-column like "LEVEL" in Oracle?

Postgres does have an equivalent to the connect by. You will need to enable the module. Its turned off by default.

It is called tablefunc. It supports some cool crosstab functionality as well as the familiar "connect by" and "Start With". I have found it works much more eloquently and logically than the recursive CTE. If you can't get this turned on by your DBA, you should go for the way Erwin is doing it.
It is robust enough to do the "bill of materials" type query as well.

Tablefunc can be turned on by running this command:

CREATE EXTENSION tablefunc;

Here is the list of connection fields freshly lifted from the official documentation.

Parameter:         Description
relname:           Name of the source relation (table)
keyid_fld:         Name of the key field
parent_keyid_fld:  Name of the parent-key field
orderby_fld:       Name of the field to order siblings by (optional)
start_with:        Key value of the row to start at
max_depth:         Maximum depth to descend to, or zero for unlimited depth
branch_delim:      String to separate keys with in branch output (optional)

You really should take a look at the docs page. It is well written and it will give you the options you are used to. (On the doc page scroll down, its near the bottom.)

Postgreql "Connect by" extension Below is the description of what putting that structure together should be like. There is a ton of potential so I won't do it justice, but here is a snip of it to give you an idea.

connectby(text relname, text keyid_fld, text parent_keyid_fld
          [, text orderby_fld ], text start_with, int max_depth
          [, text branch_delim ])

A real query will look like this. Connectby_tree is the name of the table. The line that starting with "AS" is how you name the columns. It does look a little upside down.

SELECT * FROM connectby('connectby_tree', 'keyid', 'parent_keyid', 'pos', 'row2', 0, '~')
    AS t(keyid text, parent_keyid text, level int, branch text, pos int);