What are the Google-approved methods for redirecting users to the new
location of a page?  What precautions should I take to avoid being
considered a manipulator of the index?

My site is old (many links were established mid-nineties) and I'd like
to redirect these old links in such a way that google understands that
the old links point to a new page.

1) I think I recall using the META refresh method with some success,
but presenting the user with an extra click is less than ideal.

2) What about HTTP status 301 (permanently moved)?

3) What about HTTP status 302 (temporarily moved)?

Hi there,

1) Avoid using the META refresh method - this has been abused by
spammers in the past, and might be seen as such by search engines:

"The Refresh META Tag is not a META tag which should be used on pages
you submit to search engines. Most search engines have algorithms that
will detect the Refresh tag and may either setup their system to
ignore the particular page, your entire site or may even ban your host
entirely. For this reason, we strongly suggest you do not use this
tag."
from Submit Corner
http://www.submitcorner.com/Guide/Meta/refresh.shtml

2) YES! use HTTP status 301 (see below)

3) HTTP status 302 - probably okay for redirecting your visitors for a
few days, but for search engines it is just confusing:

"302's are not a problem if it is unique content, but if you submit
both, you could make a mess on Google"
Brett_Tabke, Administrator, WebMasterWorld
http://www.webmasterworld.com/forum23/220.htm?highlight=302

Here is the official Google advice, followed by the ideas of others
via WebmasterWorld (the best place to discuss technical search engine
stuff) and some links concerning 301 redirects:
 
Change the URL of your website 
============================== 
 
Since Google's crawler associates the content of a page with its URL,
there is no way to manually change the URL that is displayed for your
website. The URL will be updated the next time we crawl your site. The
crawler revisits each site according to an automatic schedule, and we
cannot manually accelerate the date on which your site will be
recrawled.
 
If the URL of your website has changed since we last crawled it, you
may use the URL submission form and the URL removal methods described
below. However, the URL submission form does not take effect
immediately, so using the URL removal feature may leave your website
inaccessible from Google until we crawl your site again.
 
Instead of requesting a change from Google, we recommend that you ask
the sites currently linked to your old site to update their links (to
point to your new site). Also, don't forget to change any entries you
may have in the Yahoo! directory and the Open Directory. Finally, if
your old URLs redirect to your new site using HTTP 301 (permanent)
redirects, our crawler will know to use the new URL. Changes made in
this way will take 6-8 weeks to be reflected in Google.
://www.google.com/remove.html 

"I recently moved a site without any problems vis-a-vis Google. This
was both a server move and a domain name change; I moved from a "free"
web-hosting domain to a "real" .org domain with a lot more space,
features and controls for security.
 
After registering the new domain name and waiting for it to propagate,
I copied the content to the new server, got most of my incoming links
switched over to cite the new domain URLs, and got my DMOZ and other
directory listings updated. I then put a 301 (permanent) redirect on
the old server to point each old URL over to the new one, and finally
removed almost all of the content from the old server. The only
content left on that site is a "We have moved" page, .htaccess file,
and a robots.txt file...
 
...Google picked up the 301 redirects and the Open Directory update,
and quickly transferred all listings and PR over to the new site -
apparently intact. Some here on WebmasterWorld have said Google had
problems doing this, but I didn't experience any at all. But I waited
to put the 301 redirect on the old site until after I got my Open
Directory and other directory listings updated, and that may be very
important for Google."
http://www.webmasterworld.com/forum3/3806.htm 
 
Advice on this matter from a Google engineer 
http://www.searchengineworld.com/newsletter/2001/google.htm 
 
Redirecting 
http://www.webmasterworld.com/forum3/3237.htm 
 
Some info on HTTP 301: 
 
Using .htaccess 
http://www.opensourcepan.co.uk/texts/apacheredir/part2.shtml 
 
htaccess 
http://directory.google.com/Top/Computers/Internet/Web_Design_and_Development/Authoring/Webmaster_Resources/Hints_and_Tips/htaccess/
 

Search strategy: 
 
"new site" "old site" google 
://www.google.com/search?q=%22new+site%22+%22old+site%22+google&num=30
 
I trust this answers your question. If any portion of my answer is
unclear, please ask for clarification.
 
Best wishes, 
robertskelton-ga

Excellent.