I know capital expenditures show up in a company's cash-flow statement, but do they have any effect on its income statement? That is, does the income statement show only revenue-generating activities?
No, capital expenses do not directly appear on the Income Statement. There are indirect effects from CapEx that do show up (depreciation, interest if money was borrowed to fund the capex, etc.) but nothing direct.
From a T-account standpoint, A capital expense is a movement from an asset (cash) to another asset, and thus does not involve any expense or revenue accounts that would show up on the income statement.
When I observe the income statement does it show only the state of the revenue-generating activities of the company?
To be pedantic, the income statement does not show "state". It shows the change over a given time period. But yes, the income statement shows the changes due to activities that generated revenue, and the expenses that were incurred as a result of those activities.
The cost of a capital expenditure is "amortized" [~released slowly] over the useful life of the asset. Amortization is also called depreciation. This is because accounting theory tries to line up the 'expense' with the related revenue. In this case, buying an asset today not only generates revenue this year, it also generates it in future years.
ie: If a company buys a delivery van that will last 5 years, and it costs $20,000, then it might have an 'amortization expense' of $4k per year, until $20k has been amortized.
The amortization/depreciation expense translates in a very rough way, into the amount of new capital expenditure the company needs to make, in order to remain at a consistent level of operations. In the example above, if you saw $4k of amortization but $0 new capital expenditures on the cashflow statement, it would imply that the company is not buying new property to maintain the same level of operations. Of course, the amortization expense is just an estimate, and some things last far longer than the amortization period [ie: a new building might be amortized over 20 years, but might actually last 40 years before significant additional capital expenditures are required].
Like everything else with financial analysis, looking at a single number ['amortization'] tells you little. But when looking at competing factors, you learn a little more. Combining amortization with net income and current year capital expenditure spending, tells you a little more about whether the company is growing, or 'coasting'. An example from an unrelated question that was recently posted might be a car rental company. If you saw a significant amortization expense but little capital expenditures, it might imply that the fleet of rental vehicles is old and dying, so perhaps this year's net income amount will be lower next year, after cars start breaking down.