City have to face holders Bayern Munich and CSKA Moscow as well as Viktoria Plzen from the Czech Republic, but the onus is on manager Manuel Pellegrini to achieve something his predecessor Roberto Mancini did not manage - to take the club into the knockout rounds.
City's task will not be easy - among the British clubs only Chelsea could be relieved at the draw in Monaco after being chosen in Group E with Schalke, Basle and Steaua Bucharest.
Arsenal face beaten finalists Borussia Dortmund in Group F along with Rafael Benitez's Napoli and Marseille, while Manchester United will play Shakhtar Donetsk, Bayer Leverkusen and Real Sociedad in Group A.
For Celtic, there is another match-up with Barcelona in Group H, with AC Milan and Ajax making up a group of four clubs who have all previously won the European Cup.
Begiristain said Manchester City will have to show a fighting spirit.
He said: "You can never make excuses - in football never.
"It won't be easy but we have to fight. The champions Bayern are there, we also need to be careful with CSKA Moscow and also Plsen. We have to be careful with all of them."
Asked about the importance of going one better than the previous two seasons and making it to the knockout rounds, he added: "It's very important. When you start the season you have to always think you will do better than the last one.
"We have to fight for the Premier League as we love this competition. We have to do better also in the Champions League.
"There's no minimum target. We are big club, with big players and with a manager with big experience.
"We are very, very happy with our squad. We have a minimum of two very good players in each position so it's deep enough."
Chelsea have the easiest draw on paper and chief executive Ron Gourlay said the club would be boosted by Jose Mourinho's return as manager.
He said: "Jose has come back to continue the evolution of the football club. Jose went on to be very successful after he left and Chelsea continued to be too so we are now looking to continue that evolution.
"We have a very, very young squad of players now and that's very important for the next five or six years.
"We are very, very happy with the team and if the window closed today we would be fine. We have started the season very well and we are very happy with how things are going."
Gourlay added that there was no chance of Spanish midfielder Juan Mata, left out of the starting XI for both of Chelsea's games so far, being sold.
He said: "He has been our player of the year for the last two years, he is absolutely a key player in the club and he is not going anywhere."
For Celtic, there is the prospect of Barcelona's Lionel Messi and Neymar coming to Glasgow, and the Argentina striker said he is looking forward to returning to the ground where he scored a late goal in a 2-1 defeat in the group stage last season.
Messi said: "It looks an even group and there are some very nice games.
"You don't get any easy groups in the Champions League. We played Celtic last season and know AC Milan very well and Ajax have a great history.
"It's a beautiful group and I'm looking forward to some great games in the competition."
AC Milan's organising director Umberto Gandini said there was a "fatal attraction" with Barcelona - the clubs have met six times in the Champions League in the last two seasons.
He added that Milan would relish the atmosphere against Celtic.
He told Press Association Sport: "It seems as though there is a fatal attraction between Milan and Barcelona.
"We played Celtic in 2007 and we won 1-0 in overtime and we had 35-40,000 Celtic supporters right behind the goal singing from first minute to last minute and this was in San Siro.
"Celtic Park is a fantastic place and it is a great experience for the players to play there."
All four English clubs will face German opposition, and Manchester United kick off their European campaign at home to Bayer Leverkusen on Tuesday September 17, with City away at Plzen on the same night.
Chelsea's first match is at home to Swiss champions Basle on September 18 while Arsenal travel to Marseille the same night.