Hi Emilia,
Since your mother is giving money, why wouldn't you take her shopping with you?
I'd say, definitely buy some girly underwear, and some basic cosmetics (don't have to be bought in expensive mall booths, I had great advice from teenage girls working at my local Walgreens). From there, I'd go for everyday wear, androgynous style, clothes - colorful shirts, fem jeans, sneakers or no-heel sandals.
I'd suggest not to buy any breastfotms, or wigs. Wigs look kind of fake, and are too hot in summer anyway. And with breastfotms - they also don't look natural, and create a sense that you're trying to fool everyone. Just go for comfortable look, and wait till you're ready for HRT.
Well, unless you know you don't want that, then invest into some good quality forms (but don't go for too big, they are heavy, little B is still enough).
Above all, start working on your diet and exercise and learn to be positive about it

Give your parents a reason to see how your transition makes you a happier and healthier person