Having lived in Illinois, I will tell you it is a simpler formula here in Texas, but the taxes are higher for the same value. The good news is that the taxes on a similar 3 bedroom 2 bath home typically are lower than in Dupage or even Cook, despite the fact that the rates are higher here.
The taxes are normally on a combined bill, but each taxing district can issue its own bill. Every property is in multiple taxing districts, just like in Illinois. I say it's simpler because there is no assessed value versus market and a 16% or 33% multiplier and a state multiplier. Everything is market value here unless you get an exemption for being over 65 or disabled.
All owner occupants can claim an exemption for homestead. All these exemptions are either dollar amounts or a percentage of the market value. In order to calculate the taxes on a property you need to know its market value (which the tax appraisal district determines), which taxing districts it is located in (4 to 7 usually), and which exemptions apply to the property.
The unexempted rates range from about 2.4% to 3.1%. The exemptions can be as little as 0 in all districts except schools must be at least 5% for homestead. Many districts offer $5,000, 10,000 or 15,000 for homestead, and some are $50,000 for senior citizens.
Since a single formula is quite complicated because of the overlapping districts with different rates and different exemptions, if you can supply a neighborhood or address, this year's taxes can be calculated in about a minute.
In Rockwall we pay three bills, school, county and city. If you have a mortgage, the escrow pays these for you. In Dallas you can receive a bill with every taxing district on it.
When you buy a house, the value is firmly established for the next cycle, which begins January 1. Your friend's taxes may have jumped up because of a change in the value that happened because of the sale.
Say, the property was appraised by the appraisal office as $150,000 and he bought it for $175,000 last year. Then starting on January 1 this year the appraisal office would find the sales price and by May would have sent out a new appraised value notice to your friend. He can protest the value. Whatever happens to the protest, the value (probably $25k higher = +16%) will be used in September to send a bill for the whole year in October. The bill is due January next year. If your friend's taxes jumped up last October, then the change in value was due to the year prior.
Yes, as I said, compared to Illinois the tax rates are high, but the values are low -- the combination turns out to be cheaper than up there, usually.