Are there an equal number of positive and negative numbers?

If there is SOME function that gives a bijection between two sets, then these two sets are considered equally big. (Even if there is some other function between those two sets that is not onto/not 1-1). For example the set of multiples of 10 among positive integers being a proper subset of all positive integers seems to be "smaller". But we have a function $f(x)=10x$ providing bijection, and so they are considered to be of the same size (cardinality). In your case $g(x)=-x$ provides the bijection.

Yes, the existence of a one-to-one and onto mapping is exactly how equality of the size of sets (the technical term is "cardinality" is defined. The (cardinal) number of negative integers is the same as the cardinal number of positive integers, and the cardinal number of negative real numbers is the same as the cardinal number of positive real numbers.

Note however that there are more positive real numbers than positive integers, so not all infinite cardinalities are equal. The proof of that fact is done by Cantor's famous diagonal proof.

The word "infinity" is used in many places in maths and the definitions are not necessarily the same. Different symbols are used but there are still more definitions than symbols.

The most familiar symbol for infinity, $\infty$, is commonly used in calculus but it is more of a suggestive shorthand than an actual infinity. It is not normally used when discussing sizes of sets.

When discussing the size of sets, the term cardinality is usual. Clearly, normal counting won't work for infinite sets. However, as others have said, it is possible to say whether or not two sets have the same cardinality / are the same size. If a bijection (one to one and onto map) between them exists then they have the same cardinality. One counter-intuitive property of infinite sets is that it is possible that a map which is one to one and not onto exists as well one that is one to one and onto. So, the existence of a map that is one to one but not onto does not prove that one set is smaller. For that, you would also need to prove that no other map that is one to one and onto exists. The simplest example of this is the set of all natural numbers and just the even ones. Intuitively, the set of even numbers is smaller and there is an obvious map from the even natural numbers to a subset of the natural numbers. However, there is also a map between them which is one to one and onto hence they actually have the same cardinality.

By definition, a set which has a one to one and onto map to the natural numbers is called "countable". This is not countable in the day to day sense but it does mean that you could name one per second and (*), although you would not ever finish, you would name any particular one in a finite time.

(*) Assuming that you and the universe were immortal.

Many sets of intuitively different sizes are countable, for example the integers $\Bbb{Z}$, the rational numbers $\Bbb{Q}$ and the algebraic numbers $\Bbb{A}$. This cardinality / size is named countable and the symbol $\aleph_0$ is used. This is called "aleph null", "aleph naught", or "aleph zero". Aleph is the first letter of the Hebrew alphabet.

However, even though many sets which are intuitively bigger turn out to be the same size, bigger sets exist. It can be proved that there is no one to one and onto map from the natural numbers to the set of real numbers $\Bbb{R}$ so that it is bigger. Again, some apparently bigger sets are not actually bigger. For example $\Bbb{C}$, $\Bbb{R^2}$, and $\Bbb{R^3}$ are the same cardinality as $\Bbb{R}$.

Even bigger sets exist, the set of all subsets of $\Bbb{R}$ is bigger than $\Bbb{R}$. There is no biggest set.

A particularly interesting question is whether $\Bbb{R}$ is the next biggest cardinality after the countable infinity of $\Bbb{N}$. This is called the "Continuum Hypothesis". My answer is already long enough so I won't talk about that but if you are interested in this subject then you should look it up.

Finally, there is yet another type of infinity called "ordinal numbers". In this sense, the first infinity is usually written as $\omega$ and, unlike the cardinal infinities, $\omega + 1$ is different.