Not a teacher, but have a Y3 child so have gone through all this fairly recently.
There are various aspects to TTs, what are you considering? If it is helping your kids memorise them (with the aim of instant response) just like they presumably memorised number bonds, then there are various good games/apps out there that help specifically with the memorisation/instant recall thing. E.g. Squeebles TTs, TT Rockstars. Or you could get songs to listen to e.g. in the car.
At school they started with 2x, 10x and 5x and the steps towards memorising them where e.g. learning to count in twos (so being able to recite 2 4 6 8 10 12 14 16 18 20 - fluently and quickly ). Then they did lots of practice working out 2x e.g. 7, meaning they'd recite their 'counting in twos' whilst ticking off with their fingers until they reached their 7th finger. (Or they realised that x2 is the same as doubling, which they had already memorised.) Then they did 'counting in 10s' (so reciting 10 20 30 40 50 60 70 80 90 100) and the ticking off fingers thing again to work out, so 3x10 is 10 (finger one) 20 (finger 2) 30 (finger 3 - stop).
If it is more about 'what is multiplication' then there are various ways to explain. For example 'repeat addition': 3x4 is the same as adding 4 three times, so 4+4+4. You can try to make that really concrete by using items, e.g. sweets, DC1 has 4 sweets and DC2 has 4 and you have 4, so that makes 3 times 4 sweets, and we can work out how many that is by adding 4 +4+4
At school DS did lots of 'arrays' - so groups of dots, lined up in e.g. 3 rows of 4 dots each. You can then 'see' (or explain) that there are three rows, row 1 row 2 row 3, and each row has exactly the same number of dots (4). So that means 3 (rows) times 4 dots, 3x4 - and it can be 'seen' that you can work it out by adding up row1+row2+row3 = 4+4+4
These arrays also help to demonstrate/understand that 3x4 is the same as 4x3, as you can take the exact same arrray and see that there are 4 columns with three dots each. So you add up column 1+column2+column3+column4 = 3+3+3+3 = 4x3 - You just counted the dots in a different way, didn't add any or take any away, so there have got to be the same amount.
Schools differ in their approaches, some will focus most on the children gaining some understanding of what multiplication is, and introduce memorising later; others the other way around, get the kids to memorise TTs and let understanding grow later, through application.
By end of Y2 (for SATS) they are asked to demonstrate some understanding of multiplication, so a question could be '3 friends have 4 sweets each, which TWO of the following would be correct to work out how many they have altogether? 3+3+3 / 3x4 / 4+4+4 / 4+3
And they are also asked to demonstrate that they can work out (or know) the answers to 2x/5x/10x questions, e.g. 5x7=?
At the top end, they're supposed to use their understanding of multiplication as repeat addition to be able to work out a question such as 18/3 =? despite not having learned the 3x table yet. (They'd be expected to count in threes/do repeat addition of 3 until they reach 18 and see how many times they had to add 3)