I want to find out how many times a word (say foo or an IP address) occurs in a text file using the grep command on Linux or Unix-like system?

You can use the grep command to search strings, words, text, and numbers for a given patterns. You can pass the -c option to grep command. It only shows the number of times that the pattern has been matched for each file.

Show the total number of times that the word foo appears in a file named bar.txt

The syntax is:
grep -c string filename
grep -c foo bar.txt
Sample outputs:

3

To count total number of occurrences of word in a file named /etc/passwd root using grep, run:
grep -c root /etc/passwd
To verify that run:
grep --color root /etc/passwd
Pass the -w option to grep to select only an entire word or phrase that matches the specified pattern:
grep -w root /etc/passwd
OR
grep -c -w root /etc/passwd
In this example only match a word being with root:
grep --color -w '^root' /etc/passwd
grep -c -w '^root' /etc/passwd
To show only the matching part of the lines.
grep -o 'root' /etc/passwd
grep -c -o 'root' /etc/passwd
Sample session: