Store query result in a variable using in PL/pgSQL

I think you're looking for SELECT select_expressions INTO:

select test_table.name into name from test_table where id = x;

That will pull the name from test_table where id is your function's argument and leave it in the name variable. Don't leave out the table name prefix on test_table.name or you'll get complaints about an ambiguous reference.

As long as you are assigning a single variable, you can also use plain assignment in a plpgsql function:

DECLARE
    name integer;
BEGIN
    name := (SELECT t.name from test_table t where t.id = x);

Or use SELECT INTO like @mu already provided.

This works, too:

DECLARE
    name integer;
BEGIN
    name := t.name from test_table t where t.id = x;

But better use one of the first two, clearer methods, as @Pavel commented.

I shortened the syntax with a table alias additionally.
Update: I removed my code example and suggest to use IF EXISTS() instead like provided by @Pavel.

The usual pattern is EXISTS(subselect):

BEGIN
  IF EXISTS(SELECT name
              FROM test_table t
             WHERE t.id = x
               AND t.name = 'test')
  THEN
     ---
  ELSE
     ---
  END IF;

This pattern is used in PL/SQL, PL/pgSQL, SQL/PSM, ...

Create Learning Table:

CREATE TABLE "public"."learning" (
    "api_id" int4 DEFAULT nextval('share_api_api_id_seq'::regclass) NOT NULL,
    "title" varchar(255) COLLATE "default"
);

Insert Data Learning Table:

INSERT INTO "public"."learning" VALUES ('1', 'Google AI-01');
INSERT INTO "public"."learning" VALUES ('2', 'Google AI-02');
INSERT INTO "public"."learning" VALUES ('3', 'Google AI-01');

Step: 01

CREATE OR REPLACE FUNCTION get_all (pattern VARCHAR) RETURNS TABLE (
        learn_id INT,
        learn_title VARCHAR
) AS $$
BEGIN
    RETURN QUERY SELECT
        api_id,
        title
    FROM
        learning
    WHERE
        title = pattern ;
END ; $$ LANGUAGE 'plpgsql';

Step: 02

SELECT * FROM get_all('Google AI-01');

Step: 03

DROP FUNCTION get_all();

Demo: