trap reacts to the calling process signals itself. But you must call it before the signal is received. I mean, at the beginning of your script.

Furthermore, if you want to use kill -- -$$, which sends the signal also to your script, you need to clear the trap before running the kill or you will end with an infinite kill && trap loop.

For example:

#!/bin/bash
exit_script() {
    echo "Printing something special!"
    echo "Maybe executing other commands!"
    trap - SIGINT SIGTERM # clear the trap
    kill -- -$$ # Sends SIGTERM to child/sub processes
}

trap exit_script SIGINT SIGTERM

echo "Some other text"
#other commands here
sleep infinity

As explained in the comments, the problem is that the script receives the signal but is waiting for the sleep program to end before processing the received signal. So, you should kill the child processes (the sleep process in this case) in order run the trap action. You can do that with something like the following:

kill -- -$(pgrep script.sh)

Or as stated in the comments:

killall -g script.sh

On a Centos 7 test system via

$ sudo rpm -Uvh https://packages.microsoft.com/config/rhel/7/packages-microsoft-prod.rpm
$ sudo yum install dotnet-sdk-2.1

which results in dotnet-sdk-2.1-2.1.400-1.x86_64 being installed then with the test code

using System;
using System.Diagnostics;
using System.ComponentModel;
namespace myApp {
    class Program {
        static void Main(string[] args) {
            var process = new Process();
            process.EnableRaisingEvents = true; // to avoid [defunct] sh processes
            process.StartInfo.FileName = "/var/tmp/foo";
            process.StartInfo.Arguments = "";
            process.StartInfo.UseShellExecute = true;
            process.StartInfo.CreateNoWindow = true;
            process.Start();
            process.WaitForExit(10000);
            if (process.HasExited) {
                Console.WriteLine("Exit code: " + process.ExitCode);
            } else {
                Console.WriteLine("Child process still running after 10 seconds");
            }
        }
    }
}

and a shell script as /var/tmp/foo a strace stalls out and shows that /var/tmp/foo is run through xdg-open which on my system does...I'm not sure what, it seems a needless complication.

$ strace -o foo -f dotnet run
Child process still running after 10 seconds
^C
$ grep /var/tmp/foo foo
25907 execve("/usr/bin/xdg-open", ["/usr/bin/xdg-open", "/var/tmp/foo"], [/* 37 vars */] <unfinished ...>
...

a simpler solution is to simply exec a program that in turn can be a shell script that does what you want, which for .NET requires not using the shell:

            process.StartInfo.UseShellExecute = false;

with this set the strace shows that /var/tmp/foo is being run via a (much simpler) execve(2) call:

26268 stat("/var/tmp/foo", {st_mode=S_IFREG|0755, st_size=37, ...}) = 0
26268 access("/var/tmp/foo", X_OK)      = 0
26275 execve("/var/tmp/foo", ["/var/tmp/foo"], [/* 37 vars */] <unfinished ...>

and that .NET refuses to exit:

$ strace -o foo -f dotnet run
Child process still running after 10 seconds
^C^C^C^C^C^C^C^C

because foo replaces itself with something that ignores most signals (notably not USR2, or there is always KILL (but avoid using that!)):

$ cat /var/tmp/foo
#!/bin/sh
exec /var/tmp/stayin-alive
$ cat /var/tmp/stayin-alive
#!/usr/bin/perl
use Sys::Syslog;
for my $s (qw(HUP INT QUIT PIPE ALRM TERM CHLD USR1)) {
   $SIG{$s} = \&shandle;
}
openlog( 'stayin-alive', 'ndelay,pid', LOG_USER );
while (1) {
    syslog LOG_NOTICE, "oh oh oh oh oh stayin alive";
    sleep 7;
}
sub shandle {
    syslog LOG_NOTICE, "nice try - @_";
}

daemonize

With a process that disassociates itself from the parent and a shell script that runs a few commands (hopefully equivalent to your intended apt-get update; apt-get upgrade)

$ cat /var/tmp/a-few-things
#!/bin/sh
sleep 17 ; echo a >/var/tmp/output ; echo b >/var/tmp/output

we can modify the .NET program to run /var/tmp/solitary /var/tmp/a-few-things

            process.StartInfo.FileName = "/var/tmp/solitary";
            process.StartInfo.Arguments = "/var/tmp/a-few-things";
            process.StartInfo.UseShellExecute = false;

which when run causes the .NET program to exit fairly quickly

$ dotnet run
Exit code: 0

and, eventually, the /var/tmp/output file does contain two lines written by a process that was not killed when the .NET program when away.

You probably should save the output from the APT commands somewhere, and may also need something so that two (or more!) updates are not trying to be run at the same time, etc. This version does not stop for questions and ignores any TERM signals (INT may also need to be ignored).

#!/bin/sh
trap '' TERM
set -e
apt-get --yes update
apt-get --yes upgrade