The easiest and quickest way is to look up the pinyin (the original character based pinyin which is called 注音符號, which is still in common use in Taiwan). Words with the same 韻母 with the same tone are rhymes. The difficulty comes in when pinyin transcribe 注音符號 in the Latin Alphabet. For example Wu and Yu are transcribe by two different 韻母 but both end in u in the Latin Alphabet. Some words has only 聲母 but no 韻母, examples are ci, si, zi (they are rhymes if the tones are the same), the i are not the same i as in bi, di, ji, li, mi, ni, pi, qi, ti, yi, xi) which has the 韻母 i, in addition, they are not the same i as in chi, shi, zhi) which also has only 聲母 but no 韻母.
The second question concerns rhyme in modern Mandarin is harder to answer since pronunciation changes. Mandarin is adapted mainly from the dialect in BeiJing in 1932 and retained as the standard in 1949. PRC calls Mandarin PuTongHua and ROC calls Mandarin GuoYu. If you compare the dictionaries there will be minor differences PuTongHua and GuoYu went on slightly different path since 1949. So the most recent dictionary will give the correct modern Mandarin pronunciation.
A dictionary I use is 新华字典 XinHua ZiDan, the 1990 edition ISBN 7-100-00042-4/H.16 (there are probably newer one in the market now). I like this because it is organized by PinYin but also included 注音符號.