In the first cycle, you can get any of the following heart pieces: one on Clock Tower deck, one from Deku Scrub Playground, one on tree in North Clock Town, one from hand in the toilet, and one from Postman minigame. You can also get rupees to deposit into the bank and will get the Adult Wallet if you deposit 200. There are many ways to get rupees: from treasure chests, from the bushes outside Clock Town, from the Deku Scrub Playground, from the postman minigame, and from the lottery shop. You can also spend the rupees you got on a bomb bag, bombchus, Tingle maps, the treasure chest shop to get 10 deku nuts, and the Honey and Darling shop only on the final day after you get a magic meter to get a purple rupee, but can't get the heart piece from the treasure chest shop or the Honey and Darling shop. Also, if you buy bombchus in the first cycle, you won't be blocked from buying 10 bombs and 20 bombchus for 99 rupees in the second cycle.
It's pretty useless getting deku nuts in your inventory in the first cycle. Pretty much the only useful stuff you can make yourself have at the end of the first cycle is a bomb bag, bombchus in your inventory, adult wallet, rupess in the bank, or some of those 5 heart pieces. We should try and find every way of doing the first cycle such that any other way that gets everything that way of doing it gets and ends up with at least as many rupees in the bank is no faster.