One of the most difficult parts of chemistry is learning to recognize a type of reaction based solely on its reactants. This has to be done before you can apply a reaction pattern to the problem, and so it is critical to get this step right in the beginning.
In this case, you are looking at the reaction:
$$
\ce{AgNO3 + HCl -> AgCl + HNO3}
$$
You already know the products, but are questioning how these products can be formed since the activity of H is greater than Ag, implying that Ag is easier to oxidize than H. This is true - but take a look at the oxidation numbers of each species in this reaction:
$\ce{Ag: +1 -> +1}$
$\ce{NO3: -1 -> -1}$
$\ce{H: +1 -> +1}$
$\ce{Cl: -1 -> -1}$
Since there is no overall transfer of electrons, this can't be a redox reaction, and that means activities won't matter here.
That means there must be another driving force for this reaction - another reaction pattern that fits better.
I don't want to give you the answer directly, but I will give you some advice that should help:
Write the full equation - including the phases.
$$
\ce{AgNO3(aq) + HCl(aq) -> AgCl(s) + HNO3(aq)}
$$
See if you can find another reaction pattern that fits this equation better. Later, I'll update this answer to show how you can identify the correct pattern for aqueous reactions using only the reactants.
Update
As I mentioned, one of the hardest parts of chemistry is learning to identify reaction patterns based only on the reactants. This is a double-displacement (or metathesis, or precipitation) reaction. It is easy to see now, given that the reactants are aqueous and at least one of the products is solid (the precipitate). Formation of the solid is the driving force for this reaction - the quick explanation is that the forces attracting silver and chloride ions together are stronger than the solvation forces between those ions and water, as well as the forces holding them to sodium and nitrate ions.
This is good, but how do we predict that this is a precipitation reaction ahead of time?
The key is to have a good understanding of the three common types of salt/acid/base reactions that occur in aqueous solution, and to learn to recognize clues in the reactants.
Briefly, the three types that are most commonly seen are:
- Precipitation
- Acid/base (Arrhenius definition)
- Redox (single displacement)
You can recognize each of these by the reactants if you know what to look for.
Precipitation - Two soluble salts (ionic compounds), or a salt with an acid or base.
Acid/base - An Arrhenius acid and base (compound containing hydroxide)
Redox - a salt or acid and an elemental metal
There are many other types of reactions that occur in aqueous solution, and many variations of the acid/base and redox category, but these three cover the cases most commonly seen in a classroom.
Once you have identified the likely pattern that the reaction will follow, the next step is to predict the products using that pattern to see if they make sense. If they do, then you have likely chosen correctly.
You can use this algorithm for more advanced chemistry as well - in organic chemistry, for example, one of the major goals is to learn to predict reactions based on functional groups. Once you can identify functional groups and have memorized reaction patterns for them, it becomes possible to predict a huge range of reactions.
