These procedures have a long history in beginners chemistry courses. The best way is to work systematically, from HCl to remove the so-called "Group I" elements, 3 common ones with insoluble chlorides (you recall those, right?) and then separating the remaining elements into groups by precipitating them as sulfides or hydroxides, then working on the remaining groups using subtle chemical differences. If you can, to work with your example, add an excess of hydroxide to a mixture of Ba and Al, the aluminum will form a soluble complex, and you can work on them separately.
That's really the whole point of these sorts of experiments is to teach problem solving, and proper following of directions. Historically, Feigl's spot tests were developed to test for just about any ion. You can check a university library for a copy of Fritz Feigl's work. It will use the most obscure reagents and procedures, but will cover everything.
Its a good read, if you want to fill your head with this trivia. But procedures exist to analyze just about any common solution of ions, via ICP-MS instrumental analysis, and identify all practically all elements qualitatively and quantitatively to four decimal places.
FWIW, I have a book of Feigl's spot tests I bought when my local library was having a book sale. I rarely read it 'tho. Its so old, its spine is broken. I really should have it re-bound, I did buy it after all.