If you are around friends... I have the feeling they wouldn't mind if you left off the bookish-sounding grammatically correct ending of 〜たりする (something like your second example sentence.)
However, your first sentence's meaning almost sounds like this (to me):
雨が降ったら、買い物をしたり、おいしい物を食べるつもりです。
If it rains, I plan on going shopping and eating good things.
The second sentence's meaning sounds like this (to me):
雨が降ったら、買い物をしたり、おいしい物を食べたりするつもりです。
If it rains, I plan on going shopping and eating good things.
If you want to keep the context all in check, it's a good idea to end the 〜たり、〜たり pattern with する, here, though young people probably won't care... and you might find yourself in informal conversations where people break all sorts of grammatical rules. :)