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.

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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: