For all the scenes, prioritize getting your imouto's love up. Above 300 is needed to trigger the item based events. The blowjob event can be unlocked by purchasing the dirty magazine, then talking as soon as your Love is 300 or more. The bath event can be done at the same time, by buying the bath item and choosing to rest (bedroom) when love is 300.
I'm not sure when sex is unlocked, but, you have to be able to "whole body" your sister (which shows her on your lap) and be able to strip her naked. You need to fondle her until her face is blushing (almost orgasm) and then suggest sex. You'll start with missionary. This will also unlock the sex begging for $60, where doggystyle will be added automatically.
Other events from the bath have your character think that they are getting horny, with the choices to "do nothing" or "get out". Getting out will have your sister see your erection, and give you a handjob to relieve it.
Once her love exceeds a certain amount, she will no longer need to be broke for you to purchase her assistance. (She'll have sex with you (for money) even if she's "rich".)
If you're trying to save money, you can initiate sex by starting the "squeeze" ($30), making your sister horny, then suggesting sex.