We know he was born in Austria, and spent much of the pre-WW1 years in Austria and southern Germany. He served in the German army in northern France. And spent a lot of time in WW2 in East Prussia/Poland
Did he travel further? Did he, for example, visit Rome? Or any other European capitals (apart from Paris in 1940)?
All of his travels were solely made in Europe, and they frequently took place before and during the war. As a political figure, however, he was very well represented across Germany and its neighboring countries. Here's some of his major post-WW1 travels:
Hitler's first travel to Switzerland was when he travelled to Zürich to hold a speech on the 30th of August 1923, which was mainly organized by Ulrich Wille jr., Oberstkorpskommandant of the Swiss army, and Rudolf Hess, who studied at the ETH Zürich during the winter semester of 1922/1923. Hitler mainly held his speech about fighting national debt, the abolishment of the eight-hour working day and wage reductions. The main goal was to collect donations for the NSDAP, in which they succeeded, they collected between 11.000 and 123.000 Swiss francs. The visit and the speech weren't documented by the Swiss Office of the Attorney General, and Hitler denied visiting Switzerland in the Putsch process of February 1924. If these donations funded the preparations for the Putsch in November 1923 remains unclear, but the possibility that these donations massively influenced the German political landscape is given. (Source: Alexis Schwarzenbach: „Zur Lage in Deutschland“. Hitlers Zürcher Rede vom 30. August 1923, published in Traverse, Zeitschrift für Geschichte – Revue d’histoire 1/2006, p. 176–189)
Between 1923 and 1928, Hitler frequently travelled the area around Berchtesgaden privately, namely the mountainside retreat of Obersalzberg. He spent his vacations in various boarding houses. In October 1928, he rented "Haus Wachenfeld", which he bought in 1933 and renamed it "Berghof". A holiday residence at first, it later formally turned into one of Hitler's headquarters.
During his political campaign in 1932, he actively used the airplane for transportation, while his competitors relied on trains. This was propagandized as "Der Führer über Deutschland" (the Führer over Germany), and covered daily by German press. There were several "Deutschlandflüge" , which enabled Hitler to hold up to fifty speeches per week, sometimes three per day, not only in major German cities but also the German province, resulting in a huge audience, close to a million people. (Source: Ian Kershaw: "Hitler: A Biography", 2008, p. 227-228)
On June 14th 1934, Hitler travelled to the German-Italian summit which was held in Venice, where he met Mussolini for the first time. On the 3–9 May 1938, Hitler visited Rome, the only official state visit he ever did. No diplomatic negotiations were envisaged, it was a purely propagandistic visit, underlining the close relations between the two fascist nations. (Source: Bianchi Bandinelli: "Hitler e Mussolini 1938. Il viaggio del Führer in Italia", 1995)
On March 15, 1939, shortly after Germany seized control over Czechoslovakia and integrated it into the Reich as a protectorate, Hitler spent a night in the Prague Castle.
On the 23rd of March, 1939, Hitler landed on the port of Memel/Klaipėda, when it fell into German hands after the ultimatum was signed by Lithuania. He held a small speech and toured the city.
After the successful invasion of Poland, Hitler participated in the victory parade in Warsaw on October 5th, 1939.
Then, as it is widely known and already mentioned here, he visited Paris on either the 23rd or 28th of June 1940.
He travelled to Montoire-sur-le-Loir in France on the 22nd of October in 1940, where het met Pierre Laval, the French foreign minister, for an informal discussion. On 23rd October 1940, Franco and Hitler met up at Hendaye, located at the border to Spain in the Pyrénées-Atlantiques department, officially to "exchange opinions", while it was actually an attempt by Hitler to bring Franco to join the war as part of the Axis. On the 24th of October 1940, he travelled back to Montoire-sur-le-Loir to meet with Philippe Pétain, then-Chief of State of Vichy France, to discuss France's role in the new European Order.
On 26 April 1941, he visited the town of Maribor in Yugoslavia, holding a speech in which he called on his audience to "make this country German again". (Source: Geschichte und Gegenwart. Vierteljahreshefte für Zeitgeschichte, Gesellschaftsanalyse und politische Bildung. 4 (1985), Styria, p. 141)
On 4 June 1942, Hitler unofficially travelled to Imatra, Finland to congratulate Mannerheim, then commander-in-chief of Finland's defense forces, on his 75th birthday (as a side note: this is also where the only known recording of Hitler speaking in an unofficial tone came from, a Finnish engineer recorded them in a wagon).
He had two Führerhauptquartiere in both Belgium (Wolfsschlucht I, near Brûly-de-Pesche, used as HQ from 6–27 June 1940) and France (Wolfsschlucht II, near Margival, used as HQ from 16–17 June 1944). There were also two Führerhauptquartiere in Eastern Europe which were actively used by Hitler, one in Poland (Anlage Süd, near Strzyżów) and the another in the Ukraine (Werwolf, 15km north of Vinnytsia). He used the Polish one only once when he met Mussolini there, from 27–28 August 1941, while the HQ in the Ukraine was used several times, once from the 16th of July until the 30th October 1942, 19. February until the 13th of March 1943 and again on the 27th August 1943. (Source: Richard Raiber: Guide to Hitler's Headquarters, After The Battle, No. 19)