You can't count to $\omega$, you can only count towards $\omega$.
(Here, I am using $\omega$ to refer to the smallest infinite ordinal number. Don't make the mistake of confusing it with the cardinal number $\aleph_0$ or the extended real number $\infty$!)
The two counting procedures you indicate never reach $\infty$. After one step, the first procedure has reached $1$, and the second has reached $2$. After a thousand steps, the first procedure has reached $1000$, and the second has reached $2000$. After a trillion steps, the first procedure has reached $1,000,000,000,000$ and the second has reached $2,000,000,000,000$.
The point is, each step in your counting procedure is a natural number of steps away from the starting point. As such, the number you have reached at any step cannot be $\omega$!
The problem is that when you define an algorithm of the sort
- At each step, you do [something] to get to the next step
that can only get you as far as natural numbers. If you want to "count" further, or more generally talk about any other transfinite process, you need to include a second rule. A typical one is to have the steps of your transfinite process labelled by ordinal numbers, and then include a rule of the sort
- To get to any step that doesn't have an immediate predecessor, do [something else]
For example, when counting by ones, [something else] might be "move onto the smallest ordinal you haven't reached or passed yet". In that case, at step $\omega$, you have counted to $\omega$. Then at step $\omega + 1$ you've counted to $\omega + 1$ and continue on.
But when counting by twos' you might use the same [something else]. Then at step $\omega$, you've still only reached $\omega$, since prior to $\omega$ -- i.e. at the finitely-numbered steps -- you've only reached natural numbers! But then you continue counting by $2$ and at step $\omega + 1$, you've reached $\omega + 2$, then $\omega + 4$, and so forth.
Actually, there's another gotcha: when you said you were counting by $1$'s, you thought you were counting natural numbers. But $\omega$ isn't a natural number: what does it mean to count by one or by two from $\omega$? It does make sense to continue on to the next ordinal number from there to count by ones (i.e. adding by one on the right: if you care to learn arithmetic of ordinals, note that $1 + \omega = \omega \neq \omega + 1$), and skipping every other successive ordinal to count by twos. I've assumed that's what you do in the above paragraphs.
Of course, don't try to do this in the "real world" in any fashion where you have to spend one unit of time per step -- there aren't enough units of time to reach $\omega$! Other fashions are possible, though: e.g. the description of events in Zeno's paradox can be continued by setting step $\omega$ to be where Achilles has actually reached the finish line.
Another common means of passing to the transfinite is taken in geometry / analysis, by adding points "at infinity". e.g. the numbers $+\infty$ and $-\infty$ that are the endpoints of the extended real line. When we have a function defined for real numbers like $\arctan x$ or $1/x$, and the function has limits at $+\infty$, it is customary to extended the function by continuity to have the limiting value. e.g. $\arctan(+\infty) = \pi/2$ or $1/(+\infty) = 0$. But this method is mostly unrelated to discrete processes like counting or executing algorithms one step at a time.