This immediately reminded me of when I read the first volume of the "Children of the Lamp" series by P. B. Kerr. I did a little Googling just now to confirm my very vague recollections. (If you click on the series title, the link will take you to Wikipedia's detailed entry on the entire series.)
Here are some points of correspondence between your description and what I can confirm from other sources:
- When I was younger I read this fantasy series involving two siblings who find out they are wizards.
The co-stars of the series are John and Philippa Gaunt, twins who are 12 years old as the series starts.
- Their parents are involved to some extent but I think they are mostly learning from their uncle (maybe? a slightly older gentleman at
The Wikipedia entry for the series says:
During an operation to get the wisdom teeth removed, they both have
the same dream in which their uncle, Nimrod, asks that they come to
London. He tells them that they are djinn (genies).
The Wikipedia entry also confirms (as I thought I remembered) that the twins' mother tried her best to live as a normal person (i.e. zero use of magic powers) after she married her non-magical husband and settled down to raise children with him. Which explains why her brother Nimrod is the one who takes the lead in teaching the kids about magic when they're old enough to benefit from the instruction.
- There are some things that I remember specifically including: that there is one scene in which the mother tells one of the children that
they should always treat homeless people with kindness and give them
any money they could because sometimes homeless people were Angels who
would watch over you after showing selflessness;
I can't swear that I remember that bit of advice from their mother (although you may well be right), but a little Googling found this in a character entry on the Wiki dedicated to the series:
Afriel was an angel, disguised as a homeless person, who met John and
Philippa in the third book.
- in one of the books there is a detailed description of the hanging gardens of Babylon in which they are residing at the time.
I cannot swear to the "detailed description" part, but Book 2 is titled The Blue Djinn of Babylon and some Googling tells me that it does, in fact, involve a visit to the famous Hanging Gardens.
- I am pretty sure that it was a trilogy, or maybe when I read it there were only three books written, I think I read them sometime
Wikipedia says Book 1 came out in 2004, and Books 2 and 3 came out in 2006. So if you first noticed the series around late 2006, there would have been just three volumes available in bookstores and libraries at that time. (But the full series only ended when Book 7 was released in 2011.)