Which term and which counter to use when ordering simple meals? When ordering food at a cheap simple eatery in China, which words with which counters should I use that would be something vaguely like English "serve", "serving", "helping", "plate", "bowl".

For instance in English I might say "Can we have one serve of fried rice and two bowls of soup?".

In English you don't always need the words that are a bit like counter words, you can say "two fried rices", "two bowls of fried rice", or "two servings of fried rice".

Sometimes you might know that it comes on a plate or in a bowl so that might help, other times you might not be so sure, where the English words would be "helping", "serve", "serving".

But I'm assuming that in Chinese you would always need at least the counter word or both a noun and a counter word. Sometimes you might point at a picture or a menu item you can't read, or at a dish another customer has. In English you would say "one of those" but in Chinese again you would still need a measure word and probably a noun too right?
When you don't know the measure word, the safest choice is 个. It is the most common measure word, is used for things that do not have specific measure words, and can sometimes be used even if another measure word is used:

“个”作为一个量词，它的使用范围十分广泛。它可以修饰没有专用量词的名词，如人、馒头、国家、苹果等等；同时，一些有专用量词的事物，如“一只耳朵”、“一所学校”、“一家工厂”等也都可以用“个”来修饰，成为“一个耳朵”、“一个学校”、“一个工厂”，因此，有人称“个”为“万能量词”。
http://www.pep.com.cn/xgjy/hyjx/hyxxzy/hyxx/yffd/201009/t20100901_855523.htm

For food servings, 份 can be used too, but note that it is most properly used on items that are served from a larger whole. For example, this includes rice as it is cooked in one large pot and served in multiple bowls.
Where you can though, try to use the most specific measure word, as it will sound more natural:

Use 碗 for things that are served in bowls, often (plain) rice or soup noodles
Use 盘 for things served in dishes
Use 道 as a generic measure word for food dishes