How can I execute a series of commands in a bash subshell as another user using sudo?

Run a shell inside sudo: sudo bash -c 'whoami; whoami'

You can use any character except ' itself inside the single quotes. If you really want to have a single quote in that command, use '\'' (which technically is: end single-quote literal, literal ' character, start single-quoted literal; but effectively this is a way to inject a single quote in a single-quoted literal string).

You can pass the commands as standard input into sudo'ed bash with a here document:

sudo bash <<"EOF"
whoami
id
EOF

This way there is no need to fiddle with correct quoting, especially if you have multiple levels, e.g.:

sudo bash <<"EOF"
whoami
echo $USER ~
sudo -u apache bash <<"DOF"
whoami
echo $USER ~
DOF
EOF

Produces:

root
root /root
apache
apache /usr/share/httpd

(Note that you can't indent the inner terminator — it has to be alone on its line. If you want to use indentation in a here document, you can use <<- instead of <<, but then you must indent with tabs, not spaces.)

for example try this, I tested it:

sudo bash -c "cd /;ls;ls|grep o"

In this example you first change dir to /root, next list root directory and finally for root directory filter only directories having name with letter 'o'.

But i thing better way is writting script that do all you need and give exitcode for all complex action. Then you can sudo script instead group of single commands like example above.