What are the products formed when lead nitrate solution reacts with copper sulfate?

In chemistry classes, we try to make sense of the immense variety of chemical reactions by classifying reactions into groups that have certain characteristics in common. When asked to predict the product(s) of a reaction, we can often find a definitive answer by classifying the reaction and then applying the "rules" that describe how reactions in that class typically behave. Thus, our first task is to classify the reaction; then we will review what we have studied about that type of reaction and, if necessary, use information from Chemistry Reference Tables to reach a conclusion.
We are told we have lead nitrate solution reacting with copper sulfate. By this point in your course you can probably recognize that these are two ionic compounds, each made up of a metal cation and a polyatomic anion. If you have studied nomenclature, and in particular the Stock system, you know that they are properly referred to as lead (II) nitrate and copper (II) sulfate. We are told the lead nitrate is a solution, but we are not told the state of the copper sulfate.
What kind of reaction could this be? Let's run through the possibilities. We'll get combustion out of the way first. That involves something reacting with oxygen. If your chemistry course defines combustion in terms of compounds of carbon, hydrogen, and oxygen, then the products are carbon dioxide and water. Clearly, this is not combustion.
A synthesis or combination reaction involves two or more reactants combining to make a single product. The question asks about products, plural, which implies there is more than one product, so this is not a synthesis reaction.
A decomposition reaction starts with a single reactant species and results in two or more products, usually requiring the input of a good deal of energy. This starts with two reactants, so it cannot be decomposition.
A single replacement reaction involves an element reacting with a compound to produce a different element and compound. This cannot be single replacement, as we are starting with two compounds.
A double replacement reaction involves two compounds reacting to produce two other compounds. It can be an acid-base neutralization reaction or a reaction between two ionic compounds in solution in which the ions change partners. Our starting compounds are not an acid and base, but they are two ionic compounds, so this could be a double replacement reaction.
Normally we consider double replacement reactions in aqueous solution, and we look for evidence of reaction in the form of a precipitate formed or a gas given off. Typically we mix two clear (although not necessarily colorless) solutions together and pay attention to whether anything precipitates. In this problem, we are not told that the copper sulfate is aqueous; however, if we were to add solid copper sulfate to a solution containing lead nitrate, it would dissolve, as you can confirm by looking it up in a table of solubilities.
In this case, here we are predicting the products rather than doing an experiment. For a double replacement reaction, we write names of potential products by pairing the cation from each reactant with the anion from the other reactant. Once we have these names, we turn to a solubility table. If either of them is insoluble, it will precipitate from solution, and we will have a definite product. At most one product will precipitate.
In this reaction we have lead (II) nitrate and copper (II) sulfate. Potential products obtained by swapping the ions are lead (II) sulfate and copper (II) nitrate. Now, different chemistry classes use different Reference Tables with slightly different information, and you should definitely get familiar with the tables used for your class and know where to find the Solubility Rules in them. Solubility Rules are deceptively simple-looking but can trip you up if you have not practiced with them.
I have included a link to the Chemistry Reference Tables I use with my classes in North Carolina. The Solubility Rules are on page 6 (just above the guidelines for identifying different classes of reactions). We look for lead (II) sulfate first. In the left-hand column, under "Soluble," it says "all Sulfates except Calcium, Strontium, Barium, Mercury, Lead (II), and Silver." All sulfates are soluble except when paired with a short list of metal cations including lead (II). That means lead (II) sulfate is insoluble. When it is formed, it will precipitate.
I said above that there will be only one precipitate, but it is always wise to check, potentially catching our mistakes. Let's look up copper (II) nitrate as well. In the first entry in the left-hand column, under "Soluble," it says, "All nitrates . . . are soluble." While copper isn't mentioned, this means any ionic compound having any cation paired with nitrate will be soluble.
Using the solubility tables, then, we have found that lead (II) sulfate, being insoluble, will precipitate. A precipitate is a solid, so one of our products is solid lead (II) sulfate. Since your instructor called it lead sulfate, I'm sure you can, too. My North Carolina Reference Tables are specific in identifying lead (II) sulfate as insoluble, so I wanted to be clear that this is what is commonly known as "lead sulfate." The other combination, copper (II) nitrate, is soluble, so it will remain in solution. Formally, this is referred to as aqueous copper (II) nitrate. Aqueous ionic compounds exist as solvated ions in solution, so depending on the emphasis in your course you might refer to this as a solution of Cu^(2+)and NO_3^- ions.
This answer is quite long, and I'm sure your instructor will accept a much shorter answer. I thought it would be helpful to demonstrate the full process of classifying the reactions and then recalling the skills you have learned for analyzing the type of reaction once it is identified. Using this approach, you can reliably answer questions of this kind; trying to use shortcuts usually results in errors.
Lead nitrate solution, by the way, is clear and colorless. Copper sulfate solution is blue (but clear, as are all solutions). The precipitate of lead sulfate is white. Copper nitrate solution is also blue.