Unit testing beginner method

I am currently working on a project that requires us to write unit tests. This is an area where my knowledge is limited. After researching over the last few days I have seen multiple examples showing how to write basic tests that assert if a string is equal to another string.

However, what I am still unclear on is how to approach unit testing as a whole. I have an example method that I need to test below:

public static SearchResults GetSearchResults(SearchFormModel searchForm, int currentItemCount, int skip)
    {
        var results = new SearchResults();
        var client = new RestClient(Settings.Default.SearchWebServiceUrl);
        try
        {
            var request = new RestRequest("{0}", Method.GET);
            if (!searchForm.FirstName.IsNullOrEmpty()){request.AddParameter("forenames",searchForm.FirstName);}
            if (!searchForm.LastName.IsNullOrEmpty()) { request.AddParameter("surname", searchForm.LastName); }

            request.AddUrlSegment("0", "Basic");

            request.AddHeader("Accept", "application/json");

            var response = client.Execute<SearchResult>(request);
            if (response.ResponseStatus != ResponseStatus.Completed)
            {
                if (response.ErrorException != null)
                {
                    SendErrorEmail(response.ErrorException.Message);                        
                }
                throw new Exception(Settings.Default.SearchGenericError);
            }

            if (response.Data != null)
            {
                results.Valid = response.Data.Valid;
                var error = response.Data.Error;
                if (!string.IsNullOrEmpty(error)){error = Settings.Default.SearchGenericError;}
                results.Error = error;
                if (response.Data.SearchResults != null)
                {                        
                    results.SearchResults = new List<SearchResult>(response.Data.SearchResults );
                }
            }
            else
            {
                throw new Exception(Settings.Default.SearchGenericError);
            }
        }
        catch (Exception ex)
        {
            registrantList.Error = ex.Message;
        }
        return results;
    }

Would a good approach here be two break this down into two separate methods? One that sets the request and gets the response like so:

public static IRestResponse<SearchResults> GetSearchResults(SearchFormModel searchForm, int currentItemCount, int skip)
{
    var results = new SearchResult();
    var client = new RestClient(Settings.Default.SearchWebServiceUrl);
    try
    {
        var request = new RestRequest("{0}", Method.GET);
        if (!searchForm.FirstName.IsNullOrEmpty()){request.AddParameter("forenames",searchForm.FirstName);}
        if (!searchForm.LastName.IsNullOrEmpty()) { request.AddParameter("surname", searchForm.LastName); }

        request.AddUrlSegment("0", "Basic");

        request.AddHeader("Accept", "application/json");

        var response = client.Execute<SearchResult>(request);
        return response;
}

Which in turn returns to another method which should be testable ie:

public static SearchResults GetSearchResults(IRestResponse<SearchResults> response)
{
    var results = new SearchResult();
    if (response.ResponseStatus != ResponseStatus.Completed)
        {
            if (response.ErrorException != null)
            {
                SendErrorEmail(response.ErrorException.Message);                        
            }
            throw new Exception(Settings.Default.SearchGenericError);
        }

        if (response.Data != null)
        {
            results.Valid = response.Data.Valid;
            var error = response.Data.Error;
            if (!string.IsNullOrEmpty(error)){error = Settings.Default.SearchGenericError;}
            results.Error = error;
            if (response.Data.SearchResults != null)
            {                        
                results.SearchResults = new List<SearchResult>(response.Data.SearchResults );
            }
        }
        else
        {
            throw new Exception(Settings.Default.SearchGenericError);
        }
    }
    catch (Exception ex)
    {
        registrantList.Error = ex.Message;
    }
    return results;
}

Apologies for the beginner question, however I want to make sure that this approach is the way to go. Would this be the correct way to approach?

Thanks in advance