The under-ten versus open-ended rule
几 asks a question assuming a small answer. 你有几个孩子 (how many children do you have) expects an answer in single digits. 多少 drops the assumption; 你有多少钱 (how much money do you have) could be answered with any amount. In practice, 几 is the default for countable items the speaker expects to be few: family members, number of dishes, number of tries. 多少 is the default for large or uncertain quantities: prices, populations, distances.
你们家有几口人?
Nǐmen jiā yǒu jǐ kǒu rén?
How many people are in your family?
几 always needs a measure word; 多少 does not
几 must be followed by a measure word: 几个 (how many, general), 几本 (how many books), 几张 (how many sheets), 几次 (how many times). 多少 can drop the measure word with uncountable items: 多少钱 (how much money, no measure), 多少水 (how much water). When 多少 is used with countables, the measure word is optional: 多少人 and 多少个人 both work.
Age conventions
Ask a small child 你几岁 (how old are you). The 几 assumes a small number. Ask a teenager or adult 你多大 (literally "how big") or 你多大年纪 (how old are you, for polite formal questions). Asking an adult 你几岁 sounds odd and slightly babying; asking a child 你多大 is fine but less common. For elders, the respectful form is 您多大年纪 or the very formal 您贵庚.
妹妹今年几岁?哥哥多大了?
Mèimei jīnnián jǐ suì? Gēge duō dà le?
How old is the little sister this year? How old is the older brother?
Price and quantity defaults
For prices, always 多少钱, never 几钱 (which sounds archaic and wrong). For distances, 多远 or 多少公里; 几公里 would imply the answer is likely under ten kilometers, which is rarely a safe assumption. For time lengths: 几分钟 (how many minutes, for short durations), 多久 (how long, for open-ended durations), 多长时间 (how long, formal). Choose based on how precise your guess is.