How to wish people good health, etc during Chinese New Year 揮春 | 春貼 | 門聯、春仔
This is an incredibly broad question; books can be written on the subject. Because of this and I'm not an expert Chinese language user either, I'll only answer using broad generalisations (and exceptions will apply to everything I write here).

The contents

These are the words and phrases commonly used during Chinese New Year:


General well-wishes: as an important holiday, universal well-wishing themes can be used. Happiness (如意吉祥), prosperity (財源廣進), health (身壯力健), luck and success (心想事成) are all acceptable.
Chinese-specific wishes: as the festival is deeply intertwined with Chinese culture, culture-specific well-wishes are also good. Themes that are characteristic of Chinese culture include longevity (壽), harmony/serenity (四季平安) and national/cultural well-wishing (國泰民安).
CNY-specific wishes: CNY is known in Chinese as the "spring festival", so you can use the theme of spring/renewal (萬象更新); CNY is a time of family reunion so there's the theme of family/reunion (家和万事兴); according to traditional Chinese culture, everyone ages together during CNY (as opposed to the Western practice of individual birthdays), so age or birth related themes are sometimes seen; Zodiac-related phrases are also popular, for example this year is the year of the horse, so phrases with a horse (or a homonym) have been popular (马上有钱).


The forms

Chinese culture loves symmetry so the most common forms are power-of-two length phrases (4 characters, 2 characters, 8 characters). Couplets are also symmetrical in that the two lines mirror each other. Odd-numbered phrases are rare. Here are the most common forms:


Single character: by far the most common is 福 (happiness), and sometimes placed upside-down in order to invoke the phrase ‘福’倒了 / 福到了 (the ‘福’ is upside-down / happiness has arrived). Other commonly used single characters include 春 and 壽.
Four characters: four is a great number for symmetry so you see this form a lot, especially in chengyu. However, many 4-character 揮春 are not chengyu but simply phrases that are CNY-related.
Couplets: these days, couplets are most commonly used as 揮春. Basically, 春贴 are couplets that use CNY-themed phrases. There's a number of rules they have to follow which I won't cover in this answer, but the basic form is two symmetrical lines plus an optional 横批 that summarises; the 横批 is typically 4 characters.


So what should I use?

Although there are few hard and fast rules to follow, if you have to ask then you should avoid coming up with your own! It's somewhat like poetry, creative but it takes a high level of language proficiency to make good ones yourself. Here's my suggestions:


Use the most common, general phrases. These also tend to be the mass-produced ones. It may seem boring but most people do it anyway so you're in good company. I mean things like the single 福 or 恭喜發財, the ones that are never inappropriate. It's not a love letter, you don't need to impress anyone with your creativity.
Tailor to the recipient. As all well-wishing goes, personal touch almost always trumps all else. Health/longevity (长命富贵) for the elderly, profit/prosperity (生意兴隆) for entrepreneurs, growth (快高长大) for small children and so on. This is a great way to be creative, but just be careful you understand the phrases and don't commit faux pas (like 早生贵子 for unmarried/bachelors).
Find experts for bespoke phrases. Normally this is overkill, but if the occasion is very special (like a wedding, or business opening) and you have to use original phrases, get an expert. Even most native Chinese speakers find this difficult to do, or at least do it well.