I've moved this to (Non)TamaTalk - General, since it's a forums-related question as opposed to one about an issue with a virtual pet.
Anyway, whether you're on desktop or on mobile, you should see a smiley-face icon next to the text-formatting icons in the Reply box - clicking or tapping on this will pop up a list of various emoticons. TamaTalk's custom ones can be found by clicking or tapping on "Categories" and then selecting "Emoticons".
Alternatively, if you already know the "code" for one, you can type that between two colons, like so;
Code:
:tarakotchi: :lol: :Penguin:
And then it will appear in image form, like this;
Also, if you type a colon and then start typing immediately after it, a list will pop up suggesting emote matches for what you've typed, which is helpful if you've only half-remembered the correct text for a particular emote.
Once you've used a number of emotes, the list found next to the text-formatting icons will start to show a "Recently Used" heading at the very top, as well.
I hope that this is useful to yourself and to others.