What are some of the similarities between Taoism and Buddhism? Can one achieve the Tao through Buddhism and vice versa, can one achieve nirvana through Taoism?
Buddhism and Taoism are usually both judged to fall under the heading 'Perennial philosophy' because they share the same metaphysical message.
Middle Way Buddhism rejects all positive metaphysical positions. For this reason when we speak about the world we are forced to do so in riddles since we are not allowed to make or endorse a positive statement. The doctrine of Two Truths explains this and gives us a way of speaking about Reality whereby we speak of the 'self' as existing and not-existing, the equality of emptiness and fullness and so forth, and get told off by outsiders for speaking paradoxically.
In the same way Lao Tsu tells us that we cannot state the world is this or that in any case and lays down the rule 'True words seem paradoxical'. True words seem paradoxical for reasons explained by Nagarjuna in Fundamental Wisdom. Only if Reality is as Nagarjuna describes would Lao Tsu's statement be true.
So underneath the different methodology and emphasis is the same doctrine and world-view. For many people this is 'True Religion' or the Perennial philosophy.
There are, of course, Buddhists who deny Naharjuna's metaphysics but those who do must also deny the metaphysics of Lao Tsu, Plotinus, advaita, Sufism and other expressions of non-dualism. It seems to make more sense to say that truth may be discovered by anyone who seeks it and this is why so many traditions share the same underlying doctrine.
I have a nice book here somewhere by a Taoist discussing the equivalence of the message of Lao Tsu, Buddha and Jesus and will post a reference if I can find it.
So my view would be that we can achieve liberation by way of either tradition and methodology. I find Taoist ideas very useful and treat them as complementary. Taoism and Zen are like twin siblings.
PS - It would important to distinguish between Philosophical Taoism (Lao Tsu, Chung Tsu et al) and Religious Taoism, which arrived 500 years later and introduced gods, angels, heaven and hell etc. in what seems to have been an attempt to popularise the teachings. Bad idea but it gained a lot of ground and muddied the waters.
With my very little understanding of Daoism I think you can approach Tao through Buddhism but one can't achieve Theravada Nibbana via Daoism perhaps Mahayana Nibbana. I think the teaching of Daoism is very beautiful.
Daoism is hard to define, isn't it? There's the Tao Te Ching but sfaik most Daoists are Chinese, and unless you understand e.g. the Chinese language it's a bit inaccessible.
So far as I know, "one" (i.e. a sense of self, identifying with the "five aggregates" of Buddhism) does not achieve either.
I find that the doctrine that "one" is a "straw dog" (Chapter 5 of the Tao Te Ching) is suggestive. :-)
I think that "Taoism" is quite broad though, e.g. here's an aspect to it which values physical/bodily immortality (e.g. as maybe manifested in martial arts, in traditional chinese medicine, in the Taoist Immortals) which, I don't know, is maybe rather different from Buddhism.
If you've read both you may see a lot of similarities -- I quite like how the Tao Te Ching ends with hearing dogs barking but people staying home, maybe "staying home" is a metaphor for not chasing after sensuality or something like that (or maybe it's meant as purely mundane advice for a populace).
Not that literally "staying home" is necessarily a feature of Buddhism (which advocates "going forth into homelessness", also maybe different stages e.g. as illustrated in the Ten Bulls).