With the assumptions that the house in TN is held as a rental property and you are not a real estate professional, in general:
1. Investigation costs (including travel) incurred before the property was made available to rent would be capitalized as purchase costs as part of the basis of the property. Travel after the property was available to be rented would be a rental expense.
2. Furnishings with a useful life over three years can be capitalized as a rental asset (improvements) subject to depreciation. However, under the De Minimis Safe Harbor Election, you may expense any purchase of less than $2,500 that would otherwise be capitalized.
3. Improvements may be eligible for the Small Taxpayer Safe Harbor Election, which allows you to expense the cost if you meet the criteria.
In either Safe Harbor situation, you have the option to capitalize the cost instead. This would spread the cost over several years instead of creating a passive loss in the first year when your expenses may exceed your rental income.
To review the available elections for your rental property, see IRS How to make Safe Harbor Election.
For help categorizing your costs as repairs or improvements, see Table 1-1 here: IRS Pub 527 Repairs and Improvements.
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