"I like bikes."
It's easy to explain. 好き（な） is actually an adjective. Therefore, the sentence is actually "Bikes are liked by me." So, "me" is the object, but also the topic, so there is the は particle for "me" here. And "bikes" is the subject, but not the topic, so が is used. Full sentence without implying the topic: 私は自転車が好きです
Minor nitpick: in "Bikes are liked by me", "me" is not the object but rather the agent of a passive sentence (because liked is a past passive participle, not an adjective). For the purpose of this explanation, a better translation of the Japanese sentence might be "Bikes are likeable to me", where "me" is an indirect object (which would presumably be marked with に, not with を, if it wasn't also topic).
は (pronounced wa, not ha, when used as a particle) marks the topic of a sentence, what is talked about. This is a somewhat tricky concept for speakers of Western languages, which do not have any special way of marking it, and it is especially hard to learn through Duolingo since we only get isolated sentences and identifying it generally depends on what has been said before.
が (ga) marks the subject of the sentence, the one who is doing the action described by the verb, unless that subject is also the topic. If it is, は takes precedence.
を (nowadays pronounced o, not wo, in most dialects) marks the direct object, the thing or person affected by the action. Again, if it is also topic, は takes precedence.
に (ni) has several uses. It marks the indirect object, somebody who benefits from or is disadvantaged by the action. It can also mark the place where the action happens (corresponding to at/in/on), or the direction of a movement (corresponding to _to/into). In the case of an indirect object, I believe は would take precedence, but if the place or direction is topic は would be added after (because then に is not just marking a sentence constituent, but acting as a postposition).
As an example, in the sentence "Tanaka gave the book to Maria", Tanaka is the subject, the book is the direct object and Maria is the indirect object. Either can be topic, since we don't know what has been said before (are we talking about what Tanaka does, what has happened to the book or about Maria?). It should be noted, however, that this is a typical example sentence, and sentences of this form are unusual in actual English text -- in practical use, the topic (and often other nouns as well) tends to be replaced by a pronoun. But sentences like "He gave the book to Maria", "Tanaka gave it to Maria", or "Tanaka gave the book to her" become more complex to use as examples, since they are not technically equivalent. Of course, just to make things even more complicated, Japanese tends not to use pronuns in this way, but instead drops phrases alltogether if they can be implied from context.