"We need to wash the forks."
Translation:Нам надо помыть вилки.
You have нужен (m), нужна (f), нужно (n) and нужны (pl) matching gender and numerus of the subject. Here we have to use the neuter form, because нам нужно (note: 'we' in the dative case) translates to 'it is needful to us that ..." and the neutral 'it' is the so called formal subject.
Yes, this applies to all other pronouns. You can use "Надо нам..." when there is a necessity to do something, but you're not fond of it: "Надо нам и вилки, и ножи помыть, а так не хочется..." "We need to cleans both forks and knives, but ~(we don't want it so much)". "Так не хочется" is an impersonal construction, btw, and I couldn't think of a really good analog in English. :)
Hi! Please enlighten me regarding "нам" (what is this word grammarwise?) and why couldn't it be the personal pronoun "мы"? Regards from Madeira Island