I'm trying to remember this by thinking that תומכ sounds like "tummyache". And a mom will support her child when he has a tummyache. Silly but mnemonics (is that the right word?) are the only thing that help me remember the vocabulary!
it's simply how Hebrew expresses the idea of "supporting": the verb always takes ב and not את. Many verbs in other languages do this as well: it's simply a matter of learning the phrase and getting used to it, I'm afraid.