I think that a first introduction to integration should demonstrate what it can be used for in the real world, motivating it by asking how we can find the distance traveled by a train whose velocity changes over time. First we of course have to define velocity, which is the rate of change of position at the point in time, measured by taking smaller and smaller intervals of time around the point. One natural way of expressing that would end up being the derivative of position with respect to time.
Then we employ the intuition that if a train has higher velocity than another at every point in time, it would travel at least as far in the positive direction. So we can approximate the real displacement by comparing with trains that have constant velocity over each short time interval. For each of those intervals we can see that the displacement is clearly the velocity multiplied by the time elapsed, almost by definition of velocity. Subsequently we can see that if the train's velocity is sufficiently nice we can obtain good upper and lower approximations by choosing sufficiently thin intervals. Hence we get the Riemann sum.
Using this intuition, all the symbols make sense, "$\int$" being an elongated "s" for sum, "$dt$" being the size of the thin time intervals, and the bounds of the definite integral being the endpoints of the time period under discussion. Also, it is worth pointing out that using these rectangular approximations is equivalent to finding the area under the graph.
The next step is to explicitly calculate some simple examples to get a feel for what the result is like. One should then compute the displacement in terms of the time period. It is also easy to observe that like the intuition says, the displacement is additive in the time period, and so this as well as the explicitly computed examples should suggest that we should compute just the integral with a fixed lower bound, since we can obtain the general case just by a difference. This general technique of reducing the parameters needed is very useful in both mathematics and computer science.
So far we have built everything without mentioning anti-derivatives, and indeed we should not just yet. First note that the integral with a fixed lower bound can be used in all cases where the integrand is integrable to compute the integral with arbitrary bounds, even if the integrand is non-continuous. In mechanics, even though everything is quite continuous until down to the quantum level, in practice it makes sense to simplify the model of say a train that leaves a station with velocity being non-continuous. (And in other cases, acceleration. Or jerk. Or...)
Note also that the actual value of the fixed lower bound does not matter in obtaining the definite integral because the difference eliminates its effect, much as in the real world the point of reference that is used to measure displacement does not matter. Plotting the graphs obtained by using different integral lower bounds should make it clear to students that they are all the same except for a constant difference across all time, and that the constant difference corresponds to the displacement between the two chosen lower bounds.
Then the symbols "$\int f(t)\ dt$" can be said to represent simply the class of functions "$x \mapsto \int_c^x f(t)\ dt$" for all possible $c$. I find it misleading to write "$\int f(t)\ dt = \cdots + C$" because it is actually a class, not one single function. In any case, this is what the symbols should be explained to mean, and not the anti-derivative, since it may not exist. This class of functions is called the "indefinite integral", precisely because it can be used to compute arbitrary definite integrals.
But then now it is very natural to ask if there is a way to efficiently compute the indefinite integral directly without going through the approximation process. That is of course easy for certain special integrands, namely those which are continuous derivatives of some other function. The reason is that one can notice that the indefinite integral grows at a rate that is proportional to the value of the integrand at that point quite intuitively, since the thin rectangle at that point has exactly that height. A formal analysis would give the fundamental theorem of calculus, as well as the desired result that we can use anti-derivatives in the case the integrand is the continuous derivative of some other function. An integral may need to be split into separate regions before anti-derivatives can be used, and certain badly behaved cases should not be overlooked (the derivative of Volterra's function). These issues will be avoided if the indefinite integral is not confused with the anti-derivative.
Of course all these can and should be made rigorous via careful proofs at some point in time, but the above should suffice for an intuition explanation of all the basic notions. The very same ideas generalize naturally to Lebesgue integration using simple functions to approximate non-negative measurable functions, or using Lebesgue measure in place of Jordan area.