I am a level 3 rogue only proficient in light armour. Is it possible for me to become proficient in medium armour eg by multiclassing or some other means?
All the methods of gaining medium armor proficiency are based on optional rules, which is to say you need to clear them with your DM.
One method would be multiclassing, as you say, which is explained in the Player's Handbook in chapter 6. The chart on page 164 shows what you gain from multiclassing into each specific class. Quite a few of them can grant medium armor proficiency, but be aware that you have to meet the appropriate minimum stat requirements listed on the previous page.
The simpler method would be to take the "Moderately Armored" feat (p.168) when you level up to 4th, instead of your usual ability score increases.
But are you sure you really want to wear Medium armor?
As a rogue, you almost certainly have a very high Dexterity, and in that case, you really should just wear light armor. You are who it's made for. While a rogue-multiclass character might wear medium armor, it's usually because they're not focusing on Dexterity for some reason, such as because they're mostly a fighter with just a little bit of sneak attack mixed in. Rogues normally stick to light armor for a very good reason.
First off, be aware that Scale and Half Plate armor give you disadvantage on Stealth, which you really don't want, as a rogue.
Second, medium armor only allows you to add up to +2 of your dexterity bonus to your AC. As a result, with a Dex of 14 or higher, medium armor ACs run from 14 (hide) to 17 (half-plate), and you'll most likely wind up at 16 unless you're willing to drop a moderately large amount of money on half-plate (which, again, makes it hard to sneak).
Wearing studded leather armor -- which is cheaper than any medium armor except hide -- your AC is 12+Dex modifier, with no dexterity limit. If you have a starting Dexterity of 16 or 17 (+3), then your AC with studded leather would be 15. Add +2 to your Dexterity with your 4th level Ability Score Increase, and you get an AC of 16, which is as good as any medium armor other than half-plate. At 8th level, you can boost your Dexterity to 20, at which point your old studded leather armor is giving you AC 17, which is as good as anything you could get from medium armor.
If you started with Dexterity 18, you'll hit that maximum AC of 17 even faster.
What would give you a strong lift to your AC, if it really is of concern, would be acquiring proficiency with shields. At a +2 AC in exchange for your off-hand, it could be a strong benefit to you, and the methods for gaining that proficiency are essentially the same ones that could get you medium armor proficiency. If you decide to go that route, which way you want to go probably depends on whether you have an even or odd Dexterity score.
The Moderately Armored feat gives you a +1 to either Strength or Dexterity in addition to the proficiency. If you have an odd Dex score now, the +1 from the feat would bring you to the same modifier that +2 from an Ability Score Increase could get you. If your Dex is currently even, you need the full +2 to bring it up a rank, so multiclassing into Fighter (or something) is a better choice that will only slow down your Dexterity increase by 1 level rather than 4.
Multiclassing would certainly work
See Players Handbook p. 164.
Multiclassing for a single level of Cleric, Druid, Fighter, Paladin, or Ranger will give you proficiency in Medium armor.
You can also gain this proficiency by gaining a level in certain subclasses of other classes (like a Hexblade Warlock from Xanathar's Guide to Everything).
There are some other methods. Most notably the Moderately Armored Feat (PHB, p. 168), which states:
Prerequisite: Proficiency with light armor
You have trained to master the use of medium armor and shields, gaining the following benefits:
- Increase your Strength or Dexterity score by 1, to a maximum of 20.
- You gain proficiency with medium armor and shields.