How do I securely update Bitcoin Core? I am currently running a Bitcoin Core wallet 0.18.1. 
What do I need to do to update it to 0.19.0.1 version?
Would be very grateful for instructions as I am relatively new to this.
There are multiple ways in which you can update Bitcoin Core to the latest version. Each one has its own advantages and disadvantages: 


Build from source code: You can always update your software by cloning the git repository. This way you can verify the code and build the binary yourself. However, it is a resource intensive task and you will need to compile it every time a new version is released. It is also the slowest of the update options.
Update using pre-built binaries: Bitcoin Core developers build the binaries inside a deterministic build environment and sign them for verification. You can just download these binaries, verify their signature and replace the old binaries with it. This is a better, easier and the most preferred way to update.
Update using a third party PPA: You can use third party packages like the Bitcoin PPA for Ubuntu. It is the easiest method to update Bitcoin Core however, it involves trusting the launchpad and the PPA owner. Moreover, I'm seeing that the PPA does not offer the latest versions (latest release available is v0.18.0; so it's missing v0.18.1 and v0.19.0.1).


Updating using prebuilt binaries:

Visit the Bitcoin Core Download page and choose the operating system for which you want to download the pre-built binaries. After downloading the main file, click on the the verify release signature to download the signatures associated with the hosted file. Ensure that you check that the release signature is correct so as to be sure that you have downloaded the correct file. Read more about why signature verification is essential. Below are the steps for Ubuntu.

# Download the binaries for Ubuntu OS
$ wget https://bitcoincore.org/bin/bitcoin-core-0.19.0.1/bitcoin-0.19.0.1-x86_64-linux-gnu.tar.gz

# Download the signature file
$ wget https://bitcoincore.org/bin/bitcoin-core0.19.0.1/SHA256SUMS.asc

# Verify that the hash of the downloaded file matches the hash in the signature file
# This ensures you have downloaded the correct file for which the developers have signed
$ sha256sum bitcoin-0.19.0.1-x86_64-linux-gnu.tar.gz
732cc96ae2e5e25603edf76b8c8af976fe518dd925f7e674710c6c8ee5189204  bitcoin-0.19.0.1-x86_64-linux-gnu.tar.gz

$ cat SHA256SUMS.asc | grep bitcoin-0.19.0.1-x86_64-linux-gnu.tar.gz
732cc96ae2e5e25603edf76b8c8af976fe518dd925f7e674710c6c8ee5189204  bitcoin-0.19.0.1-x86_64-linux-gnu.tar.gz

# Verify the signatures. If you have not imported the signature, you would need to do so for verification
# The key is mentioned in the box: "Bitcoin Core Release Signing Keys v0.11.0+ 01EA5486DE18A882D4C2684590C8019E36C2E964"
$ gpg --recv-keys 01EA5486DE18A882D4C2684590C8019E36C2E964

# After importing the key you can now verify the release. You should get the below result.
$ gpg --verify SHA256SUMS.asc
gpg: Signature made Sun 24 Nov 2019 09:14:42 AM UTC
gpg:                using RSA key 90C8019E36C2E964
gpg: Good signature from "Wladimir J. van der Laan (Bitcoin Core binary release signing key) &lt;laanwj@gmail.com&gt;" [unknown]
gpg: WARNING: This key is not certified with a trusted signature!
gpg:          There is no indication that the signature belongs to the owner.
Primary key fingerprint: 01EA 5486 DE18 A882 D4C2  6845 90C8 019E 36C2 E964


After verifying the result, you can extract the file and copy-paste the binaries. Mostly the binaries are in /usr/bin/ or usr/local/bin file. You can check it for Bitcoin by running the command: which bitcoind.



Build from source code:

You can remove the existing binaries and build the code from source. You can follow the following commands on ubuntu:

$ git clone https://github.com/bitcoin/bitcoin.git
$ cd bitcoin
$ git checkout v0.19.0.1
$ ./autogen.sh
$ ./configure
$ make
$ sudo make install


While running the ./configure step, you can enable or disable certain features of bitcoind through the use of --enable-FEATURE and --disable-FEATURE flags. You can run ./configure --help to check the features available.



Update using third party PPA

If you have installed Bitcoin using third party PPA, then you can just run the command sudo apt-get update.