You can find the current rates on http://www.CO.Nueces.TX.US for each taxing body, but you need to know which bodies the property falls under.
The market value as determined by the appraisal district is multiplied by the corresponding rates for each of the jurisdictions the property is in, and those results added together for the tax bill. Some jurisdictions offer exemptions for the elderly, disabled, and so on. These exemptions are applied the market value for that district only. Here is a link to the rates and exemptions
When purchasing a property, the new owner pays the proration of taxes expected to be due the following January 1 by taking this years taxes and having the seller pay the portion up until closing and the buyer the rest of the year. Normally, these are used to fill the buyer's escrow account to pay taxes in January, but not more than 2 months of the total tax bill is collected from buyer at closing, while the seller's entire proration is collected.