Viability of proposed carbon-carbon bond-forming reaction
With some some minor modifications, your two stage reaction

Kolbe nitrile synthesis
alkaline hydrolysis of the nitrile to corresponding carboxylic acid

will work in principle.
In the Organikum, 21. edition, 2001, p. 256ff, a general procedure for the reaction of alkyl bromides with sodium cyanide is provided.
The procedure distinguishes between reactive and less reactive halides.
Reactive halides are benzyl halides (unsubstituted, methoxy- or methyl-substituted.
Less reactive halides are halogen-substituted benzyl halides and plain alkyl halides.
For the latter, the following procedure is described:

In einem 1-l-Dreihalskolben mit Rührer, Rückflußkühler und Innenthermometer werden 250 ml Triethylenglycol, 1,25 mol gut gepulvertes, trockenes Natriumcyanid und l mol Alkylbromid oder Alkylchlorid unter gutem Rühren vorsichtig erhitzt. Der Beginn der stark exothermen Reaktion ist bei den niederen Alkylhalogeniden daran zu erkennen, daß die Lösung aufsiedet. Man steigert die Temperatur langsam auf 14O °C, bei der Umsetzung von Benzylhalogeniden nur auf 100 °C, und rührt bei dieser Temperatur noch 30 Minuten.

(Translation: In a 1-liter three-necked flask equipped with stirrer, reflux condenser and internal thermometer, 250 ml of triethylene glycol, 1.25 mol of well-powdered and dry sodium cyanide and 1 mol of alkyl bromide or alkyl chloride are carefully heated. The beginning of the strongly exothermic reaction in the lower alkyl halides is evident from the fact that the solution settles. The temperature is slowly raised to 140 °C, while the reaction of benzyl halides is only to 100 °C, and stirring is continued at this temperature for 30 minutes.)
The solvent is triethylene glycol with a boiling point of 291 °C, and the reaction is performed under exclusion of water using dry sodium cyanide.
As I have mentioned in my comment, the low boiling point of bromoethane is a disadvantage!
In fact, the Organikum recommends the use of diethyl sulfate (bp 209 °C) for the synthesis of propionitrile (bp 118 °C) and suggests to isolate the product by distilling it off the reaction mixture.
For further purification of the more volatile nitriles with an alkyl chain shorter than $\ce{C5}$, it is mentioned to

wash then with brine
dry over calcium chloride
re-distill over a 30 cm Vigreux column

For this reaction (to propionitrile), the yield is given as 90%. (For valeronitrile, hexanenitrile and heptanenitrile, which all can be prepared from the corresponding bromides, the given yields are around 80%).
According to the Organikum (p. 500) the hydrolysis of primary nitriles is achieved by heating 1 mol of the nitrile with 2 mol of 25% aqueous sodium hydroxide solution under reflux until no more ammonia evaporates (2 to 1o hours).
Whether you really want to use this two stage procedure of perform a Grignard reaction of the bromides with $\ce{CO2}$ is up to you. As mentioned in the comments, the advantage of the Grignard pathway is that it's pretty much a one-pot reaction all the way down to the carboxylic acid. The downside is the requirement of a dry solvent.

Update
In a comment, you asked for a recommendation on which carboxylic acid to synthesize.
I would synthesize a compound if

I need it (for further studies)
it is not commercially available
it costs a fortune. (Make sure that the halide that you need as a starting material is not more expensive than the final carboxylic acid.)

If it is just for practise, I'd go for phenylacetic acid (or a donor-substituted derivative), which has the following advantages:

benzyl bromide is a (weak) lacrymator, but not a CMR compound
the reaction with sodium cyanide can be carried out in refluxing acetone (instead of triethylene glycol)
phenylacetic acid, the final product, has a melting point of 77 °C and a rather low solubility in water (&lt; 2 g/L). The product can be purified by crystallization.